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# Download I. Course Title Advanced Placement Physics C: Mechanics and

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I. Course Title Advanced Placement Physics C: Mechanics and Electricity & Magnetism II. Course Description The AP Physics C course at TJ is actually two distinct, one-semester courses. These two courses at TJ are intended to be the equivalent of the first two semesters of the university sequence in physics for scientists’ or engineers. The three-semester or two-year sequence of lower-division physics courses at a major university always includes classical mechanics as one semester and classical electricity & magnetism as another semester with additional semesters covering modern physics and optics. Both courses are strongly calculus-based, and are explicitly aimed at those intending to major in engineering or the physical sciences. There is a significant laboratory component to both courses. Students study a mathematically substantial formulation of Newtonian mechanics (first semester) and electricity and magnetism (second semester), including vector and calculusbased treatment of particle kinematics (motion), Newton’s interaction model, energy, linear momentum, angular momentum, systems of particles, oscillators, and Newtonian gravity in the first semester. Topics covered in the second semester include electromagnetic fields, superposition, electrostatics, magnetostatics, induction, electric currents and elementary circuits, Maxwell’s equations in integral form and the Lorentz force law. Students are thoroughly prepared to take both the Mechanics and Electricity and Magnetism sections of the Advanced Placement Physics C examination however preparing students for these exams is not the direct purpose of this course, rather a side outcome. It is strongly recommended that the AP Physics C course at TJ be taken as a second year Physics course. A first year course (Honors Physics) aimed at developing a through understanding of important physical principles and that permits students to explore concepts in the laboratory should be taken and will provides a richer experience in the process of science and better prepares them for the more analytical approaches taken in the AP Physics C course. Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 III. TJ Specific Performance Indicators Mechanics Course Kinematics Standard 1: Motion in One Dimension Benchmark 1.a Students should understand the general relationships among position, velocity, and acceleration for the motion of a particle along a straight line. Indicator 1.a.1 Given a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, they can recognize in what time intervals the other two are positive, negative, or zero and can identify or sketch a graph of each function of time. Indicator 1.a.2 Given an expression for one of the kinematic quantities, position, velocity or acceleration, as a function of time, they can determine the other two as a function of time, and find when these quantities are zero or achieve their maximum and minimum values. Indicator 1.a.3 Given multi-dimensional, time dependent vector functions for one of the kinematics quantities students can determine all other kinematic quantities by the explicit use of calculus techniques including chain rule, product rule, or various forms of integration of polynomial and non-polynomial functions. Benchmark 1.b Students should understand the special case of motion with constant acceleration Indicator 1.b.1 Write down expression for the velocity and position as functions of time, and identify or sketch graphs of these quantities. Indicator 1.b.2 Use the equations of constant acceleration singularly or in combination to solve problems involving one-dimensional motion with constant acceleration. Indicator 1.b.3 Use the equations of constant acceleration singularly or in combination as vector valued functions involving one-dimensional motion with constant acceleration. Mechanics Course Kinematics 2 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Benchmark 1.c Students should know how to deal with situation in which acceleration is a specified function of velocity and time so they can write an appropriate differential equation and solve for other kinematic time-varying variables. Indicator 1.c.1 Using separation of variables and integration solve first order ordinary differential equations in time. Indicator 1.c.2 Using separation of variables and integration solve second order ordinary differential equations in time. Indicator 1.c.3 Using separation of variables, chain rule, and integration solve first order ordinary differential equations where kinematic variables vary in terms of position. Standard 2: Motion in Two and Three Dimensions Benchmark 2.a Students should be able to add, subtract, and resolve displacement, velocity, and acceleration vectors in two or more dimensions. Indicator 2.a.1 Determine components of a vector along two or more mutually perpendicular axes. Indicator 2.a.2 Determine the net displacement of a particle or the location of one particle relative to another including particles that are not moving in a straight line. Indicator 2.a.3 Determine the change in velocity of a particle or the velocity of one particle relative to another including particles that are not moving in a straight line. Indicator 2.a.4 Determine the change in acceleration of a particle of the acceleration of one particle relative to another including particles that are not moving in a straight line. Indicator 2.a.5 Given vector functions of time in two or more dimension which describe the motion of two or more particles determine vector expressions for the relative position, velocity, and acceleration of the particles with respect to each other. Mechanics Course Kinematics 3 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Benchmark 2.b Students should understand the motion of projectiles in a uniform gravitational field, so they can Indicator 2.b.1 Write down vector expressions for the horizontal and vertical components of position, velocity, and acceleration as functions of time, and sketch or identify graphs of these components. Indicator 2.b.2 Use vector expressions for the position, velocity, and acceleration of a projectile that is projected at arbitrary angle and initial velocity to analyze the motion of the projectile. Indicator 2.b.3 Use calculus to analyze the rates of change of position, velocity, and acceleration vectors during projectile motion to determine maximums, minimums, and other quantities. Benchmark 2.c Students should be able to analyze the motion of particles moving in circles or other nonlinear cases Indicator 2.c.1 Using calculus or other mathematical techniques determine static or time dependent vector expressions for position, tangential velocity, centripetal acceleration, tangential acceleration, and total acceleration of a moving particle. Standard 3: Vector Algebra Benchmark 3.a Students should be able to make use of and solve problems using vector algebra Indicator 3.a.1 Use vector dot products to determine the projection of one vector onto another vector in Cartesian systems. Indicator 3.a.2 Use vector cross products to determine the product of two vectors in Cartesian systems. Indicator 3.a.3 Make use of vector identities such at BAC-CAB rule to simplify vector expressions Indicator 3.a.4 Make use of vectors in Polar, Spherical, or Rotating coordinate systems including unit vectors in these systems. Mechanics Course Kinematics 4 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Newtonian Laws of Motion Standard 4: Static Equilibrium and the First Law Benchmark 4.a Students should be able to analyze situation in which a particle remains at rest, or moves with constant velocity. Indicator 4.a.1 Use Free Body Diagrams to analyze the static motion of particles including the resolving of multiple force vectors into arbitrary orthogonal axis. Standard 5: Dynamics of a single particle and the Second Law Benchmark 5.a Students should understand the relation between the forces that act on an object and the resulting changes in the objects velocity. Indicator 5.a.1 Calculate, for an object moving in one dimension the velocity change that results when a constant force acts over a specified time interval. Indicator 5.a.2 Calculate, for an object moving in one or more dimensions the vector or scalar velocity change that results when a time dependent vector force acts over a specified time interval. Indicator 5.a.2 Determine, for an object moving in a plane whose velocity vector undergoes a specified change over a specified time interval, the average force that acted on the object. Benchmark 5.b Students should understand how Newton’s Second Law applies to an object subject to forces such as gravity, the pull of strings, or contact forces. Indicator 5.b.1 Draw a well-labeled Free Body Diagram showing only real forces that act on the object. Indicator 5.b.2 Write down the vector equation that results from applying Newton’s Second Law to the object, and take components of this equation along appropriate axis. Indicator 5.b.3 Analyze situations in which an object moves with specified acceleration under the influence of one or more forces to determine the magnitude and direction of the net force, or one of the forces that make up the net force. Mechanics Course Newtonian Laws of Motion 5 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Benchmark 5.c Students should understand the significance of the coefficient of friction Indicator 5.c.1 Write down the relationship between the normal and frictional forces on a surface. Indicator 5.c.2 Analyze situation in which an object moves along a rough inclined plane or horizontal surface. Indicator 5.c.3 Analyze under what circumstances an object will star to slip, or to calculate the magnitude of the force of static friction. Indicator 5.c.4 Understand the complicated molecular and atomic origins of the frictional force and relate this physical property back to core Chemistry concepts. Indicator 5.c.5 Analyze the motion of objects under position dependent frictional forces using differential equations and methods of integration. Benchmark 5.d Students should understand the effect of drag forces on the motion of an object. Indicator 5.d.1 Find the terminal velocity of an object moving vertically under the influence of a retarding force that depends on velocity. Indicator 5.d.2 Describe qualitatively, with the aid of graphs, the acceleration, velocity, and displacement of a particle under the influence of a velocity dependent drag force when it is released from rest or is projected vertically with specified initial velocity. Indicator 5.d.3 Using Newton’s Second Law, write differential equations for the velocity of an object as a function of time if the object is moving under the influence of a velocity dependent drag force. Indicator 5.d.4 Using the method of separation of variables derive the equations of motion from differential equations for objects moving under the influence of a velocity dependent drag force and a constant force. Mechanics Course Newtonian Laws of Motion 6 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Standard 6: Systems of two or more objects and the Third Law Benchmark 6.a Students should understand Newton’s Third Law and how it applies to the interactions of particles and objects. Indicator 6.a.1 Identify force pairs and the objects on which the act and state the magnitude and direction of each force. Indicator 6.a.2 Apply Newton’s Third Law in analyzing the force of contact between two objects that accelerate together along a horizontal or vertical line, between two surfaces that slide across one another, of two objects moving in circular paths. Indicator 6.a.3 Know that Tension is constant in a light string that passes over a massless pulley and should be able t use this fact in analyzing the motion of a system of two objects joined by a string. Indicator 6.a.4 Know that Tension is not constant in massive strings or similar objects and apply calculus techniques to analyze systems such as this. Indicator 6.a.5 Solve problems in which applications of Newton’s Laws lead to two or more simultaneous linear equations involving unknown forces or accelerations. Work, Energy, and Power Standard 7: Work and the Work-Energy Theorem Benchmark 7.a Students should understand the definition of work, including when it is positive, negative, or zero. Indicator 7.a.1 Calculate the work done by a specified constant force on an object that undergoes a specific displacement. Indicator 7.a.2 Relate the work done by a force to the area under a graph of force as a function of position, and calculate this work in the case the force is both linear and non-linear. Indicator 7.a.3 Use integration to calculate the work performed by a force that varies with position on an object that undergoes a specified displacement in one or multiple dimensions. Indicator 7.a.4 Use the scalar product operation to calculate the work performed by a specified force on an object that undergoes a displacement in a plane. Mechanics Course Work, Energy, and Power 7 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Benchmark 7.b Students should understand and be able to apply the work-energy theorem. Indicator 7.b.1 Calculate the change in kinetic energy or speed that results from performing a specified amount of work on an object. Indicator 7.b.2 Calculate the work performed by the net force or by each of the forces that make up the net force on an object that undergoes a specified change in speed or kinetic energy. Indicator 7.b.3 Apply the theorem to determine the change in an object’s kinetic energy and speed that results from the application of specified forces, or to determine the force that is required to bring and object to rest in a specified distance. Standard 8: Forces and Potential Energy Benchmark 8.a Students should understand the concept of a conservative force. Indicator 8.a.1 State alternative definitions of conservative force and explain why these definition are equivalent. Indicator 8.a.b Describe examples of conservative forces and non-conservative forces. Benchmark 8.b Students should understand the concepts of potential energy. Indicator 8.b.1 State the general relation between force and potential energy, and explain why potential energy can be associated only with conservative forces. Indicator 8.b.2 Calculate a potential energy function associated with a specified one-dimensional force. Indicator 8.b.3 Calculate the magnitude and direction of a one-dimensional force when given the potential energy function in terms of position for the force. Indicator 8.b.4 Calculate the turning points of objects trapped in potential wells via the magnitude and direction of the associated force. Indicator 8.b.5 Write an expression for the force exerted by an ideal spring and for the potential energy of a stretched or compressed spring including non-linear springs. Mechanics Course Work, Energy, and Power 8 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 8.b.6 Calculate the potential energy of one of more objects in a uniform gravitational field. Standard 9: Conservation of Energy Benchmark 9.a Students should understand conservation of energy. Indicator 9.a.1 Identify situation in which mechanical energy is or is not conserved. Indicator 9.a.2 Apply conservation of energy in analyzing the motion of systems of connected objects, such as an Atwood machine. Indicator 9.a.3 Apply conservation of energy in analyzing the motion of objects that move under the influence of springs. Indicator 9.a.3 Apply conservation of energy in analyzing the motion of objects that move under the influence of other non-constant one-dimensional forces. Indicator 9.a.4 Solve complex problems that incorporate multiple concepts of conservation or energy, work energy theorem, and make use of calculus techniques. Indicator 9.a.5 Recognize and solve problems that call for application both of conservation of energy and Newton’s Laws. Benchmark 9.b Students should understand the definition of Power. Indicator 9.b.1 Calculate the power required to maintain the motion of an object with constant acceleration (e.g., to move on object along a level surface, to raise and object at a constant rate, or to overcome friction for an object that is moving at constant speed). Indicator 9.b.2 Calculate the work performed by a force that supplies constant power, or the average power supplied by a force that performs a specified amount of work. Indicator 9.b.3 Analyze a time or position dependent power input to an object using differential equations to calculate the time dependent equations of motion. Mechanics Course Work, Energy, and Power 9 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Systems of particles and linear momentum Standard 10: Center of mass Benchmark 10.a Students should understand the technique for finding center of mass and its connections to linear momentum and force. Indicator 10.a.1 Identify by inspection the center of mass of a symmetrical object. Indicator 10.a.2 Locate the center of mass of a system consisting of two such objects. Indicator 10.a.3 Use integration to find the center of mass of symmetric objects with constant and nonuniform densities. Use of non-Cartesian coordinate systems is also expected. Indicator 10.a.4 Apply the relation between center-of-mass velocity and linear momentum, and between center-of-mass acceleration and net external force for a system of particles. Indicator 10.a.5 Define center of gravity and to use this concept to express the gravitational potential energy of a rigid object in terms of the position of its center of mass. Standard 11: Impulse and Momentum Benchmark 11.a Students should understand impulse and linear momentum Indicator 11.a.1 Relate mass, velocity, and linear momentum for a moving object, and calculate the total linear momentum of a system of objects. Indicator 11.a.2 Relate impulse to the change in linear momentum and the average force acting on an object. Indicator 11.a.3 State and apply the relations between linear momentum and center-of-mass motion for a system of particles Indicator 11.a.4 Calculate the area under a force verses time graph and relate it to the change in momentum of an object. Indicator 11.a.5 Calculate the change in momentum of an object given a time dependent function for the net force action on the object. Mechanics Course Systems of particles and linear momentum 10 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Standard 12: Conservation of Linear Momentum and Collisions Benchmark 12.a Students should understand linear momentum conservation. Indicator 12.a.1 Explain how linear momentum conservation follows as a consequence of Newton’s Third Law for an isolated system. Indicator 12.a.2 Identify situations in which linear momentum, or component of the linear momentum vector, is conserved. Indicator 12.a.3 Apply linear momentum conservation to one-dimensional elastic and inelastic collisions and two-dimensional completely inelastic collisions. Indicator 12.a.4 Apply linear momentum conservation to two-dimensional elastic and inelastic collisions. Indicator 12.a.5 Analyze situations in which two or more objects are pushed apart by a spring or other agency, and calculate how much energy is released in such a process. Indicator 12.a.6 Using differential equations, describe the motion of a rocket using the concepts of impulse and conservation of momentum. Benchmark 12.b Students should understand frames of reference Indicator 12.b.1 Analyze the uniform motion of an object relative to a moving medium such as a flowing stream. Indicator 12.b.2 Analyze the motion of particles relative to a frame of reference that is acceleration horizontally or vertically at a uniform rate. Indicator 12.b.3 Model and analyze one-dimensional collisions using the center of mass reference frame. Circular motion and rotation Standard 13: Uniform circular motion Benchmark 13.a Students should understand the uniform circular motion of a particle. Mechanics Course Circular motion and rotation 11 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 13.a.1 Prove the relationship between radius of a particle moving in a circular path and the speed of revolution to the centripetal acceleration. Indicator 13.a.2 Describe the direction f the particle’s velocity and acceleration at any instant during the motion. Indicator 13.a.3 Determine the components of the velocity and acceleration vectors at any instant and sketch or identify graphs of these quantities. Indicator 13.a.4 Analyze situations in which an object moves with specified acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force, or of one of the forces that make up the net force, in situation such as 1) Motion in a horizontal circle (e.g. mass on a rotating merry-go-round, or car rounding a banked curve) or 2) Motion in a vertical circle (e.g. mass swinging on the end of string, cart rolling down a curved track, rider on a Ferris wheel) Benchmark 13.b Students should understand the concept of torque and rotational statics. Indicator 13.b.1 Calculate the magnitude and direction of the torque associated with a given force. Indicator 13.b.2 Calculate the torque on a rigid object due to gravity. Indicator 13.b.3 State the conditions for translational and rotational equilibrium of a rigid object. Indicator 13.b.4 Apply these conditions in analyzing the equilibrium of a rigid object under the combined influence of a number of coplanar forces applied at different locations. Benchmark 13.c Students should develop a qualitative and quantitative understanding of rotational inertia Indicator 13.c.1 Determine by inspection which set of symmetrical objects of equal mass has the greatest rotational inertia. Indicator 13.c.2 Determine by what factor an object’s rotational inertia changes if all its dimensions are increased by the same factor. Indicator 13.c.3 Determine the moment of inertia for a collection of point masses lying in a plane about an axis perpendicular to the plan. Mechanics Course Circular motion and rotation 12 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 13.c.4 Using integration determine the moment of inertia of symmetric objects with constant and non-uniform densities about an arbitrary axis perpendicular to the plane. Indicator 13.c.5 Using Students should be able to state, prove, and apply the parallel-axis theorem. Benchmark 13.d Students should understand and be able to apply rotational kinematic concepts. Indicator 13.d.1 Students should understand the analogy between translational and rotational kinematics so they can write and apply relations among the angular acceleration, angular velocity, and angular displacement of an object that rotates about a fixed axis with constant angular acceleration. Indicator 13.d.2 Students should be able to use the right-hand rule to associate an angular quantities and their associated vectors with a rotating object. Indicator 13.d.3 Students should make use of the vector cross product to calculate the direction of the angular quantities. Indicator 13.d.4 Describe in detail the analogy between fixed-axis rotation and straight-line translation. Indicator 13.d.5 Determine the angular acceleration with which a rigid object is accelerated about a fixed axis when subjected to a specified external torque. Indicator 13.d.6 Determine the radial and tangential acceleration of a point on a rigid object. Indicator 13.d.7 Apply conservation of energy to problems of fixed-axis rotation. Indicator 13.d.8 Analyze problems involving string and massive pulleys. Standard 14: Angular momentum and its conservation Benchmark 14.a Students should be able to use the vector product and the right-hand rule Indicator 14.a.1 Calculate the torque of a specified force about an arbitrary origin. Mechanics Course Circular motion and rotation 13 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 14.a.2 Calculate the angular momentum vector for a moving particle. Indicator 14.a.2 Calculate the angular momentum vector for a rotating rigid object in simple cases where this vector lies parallel to the angular velocity vector. Benchmark 14.b Students should understand angular momentum conservation Indicator 14.b.1 Recognize the conditions under which the law of conservation is applicable and relate this law to one- and two- particle systems such as satellite orbits. Indicator 14.b.2 State the relations between net external torque and angular momentum, and identify situations in which angular momentum is conserved. Indicator 14.b.3 Analyze problems in which the moment of inertia of an object is changed as it rotates freely about a fixed axis. Indicator 14.b.4 Analyze a collision between a moving particle and a rigid object that can rotate about a fixed axis or about its center of mass. Oscillations and gravitation Standard 15: Simple harmonic motion Benchmark 15.a Students should understand simple harmonic motion and apply that knowledge to simple systems like a mass on a spring. Indicator 15.a.1 Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period and frequency of the motion. Indicator 15.a.2 Write down an appropriate expression for sinusoidal displacement. Indicator 15.a.3 Find an expression for velocity as a function of time. Indicator 15.a.4 State the relations between acceleration, velocity, and displacement, and identify point in the motion where these quantities are zero or achieve their greatest positive and negative values. Indicator 15.a.5 State and apply the relation between frequency and period. Mechanics Course Oscillations and gravitation 14 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Indicator 15.a.6 Last Revised July 2016 మ Prove that a system that obeys a differential equation of the form మೣୀିఠమ௫ must execute simple harmonic motion, and determine the frequency and period of such motion. Apply the results to physical systems. Indicator 15.a.7 State how the total energy of an oscillating system depends on the amplitude of the motion, sketch, or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic. Indicator 15.a.8 Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic and potential energy is constant. Indicator 15.a.9 Calculate the maximum displacement or velocity of a particle that moves in simple harmonic motion with specified initial position and velocity. Indicator 15.a.10 Develop a qualitative understanding of resonance so they can identify situation in which a system will resonate in response to a sinusoidal external force. Indicator 15.a.11 Derive and apply the expression for the period of oscillation of a mass on a spring. Indicator 15.a.12 Analyze problems in which a mass hangs from a spring and oscillates vertically or problems in which a mass is attached to a spring oscillates horizontally. Indicator 15.a.13 Determine the period of oscillation for systems involving series or parallel combination of identical springs, or springs of differing lengths. Benchmark 15.b Students should be able to apply their knowledge of simple harmonic motion to the case of a simple and physical pendulum. Indicator 15.b.1 Derive the expression for the period of a simple pendulum. Indicator 15.b.2 Apply the expression for the period of a simple pendulum. Indicator 15.b.3 Apply Maclaurin series expansions to reduce the order of modeling equations to first order (linear) terms, which can be used to derive the period of small oscillations Indicator 15.b.2 Derive an expression for the period of a simple physical pendulum from first principles making use of moments of inertia and an understanding of rotational dynamics. Mechanics Course Oscillations and gravitation 15 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Standard 16: Newton’s Law of Gravity Benchmark 16.a Students should know Newton’s Law of Universal Gravitation. Indicator 16.a.1 Determine the force that one spherically symmetrical mass exerts on another. Indicator 16.a.2 Determine the force that one spherically symmetrical mass of non-uniform density exerts on another. Indicator 16.a.3 Determine the strength of the gravitational field at a specified point outside a spherically symmetrical mass. Indicator 16.a.4 Determine the strength of the gravitational field at a specified point outside a nonspherically symmetrical mass with non-uniform density. Indicator 16.a.5 Using Shell Theorem, calculate the gravitational force or field strength inside and outside uniform spheres as well as spheres with non-uniform mass density. Benchmark 16.b Students should understand the motion of an object in orbit under the influence of gravitational forces. Indicator 16.b.1 For a circular orbit recognize that the motion does not depend on the object’s mass; describe quantitatively how the velocity, period of revolution, and centripetal acceleration depend upon the radius of the orbit; and derive expressions for the velocity and period of revolution in such an orbit. Indicator 16.b.2 Derive Kepler’s Third Law for this case of circular orbits. Indicator 16.b.3 Derive and apply the relations among kinetic energy, potential energy, and total energy for circular orbits. Indicator 16.b.4 For a general orbit derive Kepler’s second law and use all Laws to quantitatively describe the motion of objects in elliptical orbits. Indicator 16.b.5 Apply conservation of angular momentum to determine the velocity and radial distance at any point in a general elliptical orbit. Mechanics Course Oscillations and gravitation 16 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 16.b.6 Apply angular momentum conservation and energy conservation to relate the speeds of an object at the two extremes of an elliptical orbit. Indicator 16.b.7 Apply energy conservation in analyzing the motion of an object that is projected straight up from a planet’s surface or that is projected directly towards the planet from far above the surface. Electricity and Magnetism Course Electrostatics Standard 17: Charge and polarization Benchmark 17.a Students should understand the concept of electric charge and charge polarization. Indicator 17.a.1 Describe the types of charge and the attraction and repulsion of charges. Indicator 17.a.2 Describe polarization and induced charges. Benchmark 17.b Students should understand Coulomb’s Law and the principle of superposition Indicator 17.b.1 Calculate the magnitude and direction of the force on a positive or negative charge due to other specified point charges. Indicator 17.b.2 Analyze the motion of a particle of specified charge and mass under the influence of an electrostatic force. Standard 18: Electric field and electric potential Benchmark 18.a Students should understand the concept of electric field. Indicator 18.a.1 Define electric field in terms of the force on a test charge. Indicator 18.a.2 Describe and calculate the electric field of a single point charge. Electricity and Magnetism Course Electrostatics 17 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 18.a.3 Calculate the magnitude and direction of the electric field produced by two or more point charges. Indicator 18.a.4 Using Maclaurin and other expansions or approximations determine the near and far field approximations from two or more point charges. Indicator 18.a.5 Calculate the magnitude and direction of the force on a positive or negative charge placed in a specified field. Indicator 18.a.6 Interpret an electric field diagram. Indicator 18.a.7 Analyze the motion of a particle of specified charge and mass in a uniform electric field. Benchmark 18.b Students should understand the concept of electric potential. Indicator 18.b.1 Determine the electric potential in the vicinity of one or more point charges. Indicator 18.b.2 Calculate the electrical work done on a charge or use conservation of energy to determine the speed of a charge that moves through a specified potential difference. Indicator 18.b.3 Determine the direction and approximate magnitude of the electric field at various positions given a sketch of equipotentials. Indicator 18.b.4 Calculate the potential difference between two points in a uniform electric field, and state which point is at the higher potential. Indicator 18.b.5 Calculate how much work is required to move a test charge from one location to another in the field of fixed point charges. Indicator 18.b.6 Calculate the electrostatic potential energy of a system of two or more point charges, and calculate how much work is required to establish the charge system. Indicator 18.b.7 Use integration to determine electric potential difference between two points on a line, given electric field strength as a function of position along that line. Indicator 18.b.8 State the general relationship between field and potential, and define and apply the concept of a conservative electric field. Electricity and Magnetism Course Electrostatics 18 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 18.b.9 Define the relationship between field and potential as a vector gradient operator and apply this relationship to calculate the electric field vector in a two dimensional potential. Standard 19: Gauss’ Law Benchmark 19.a Students should understand the relationship between electric field and electric flux. Indicator 19.a.1 Calculate the flux of an electric field through an arbitrary surface or of a field uniform in magnitude over a Gaussian surface and perpendicular to it. Indicator 19.a.2 Calculate the flux of the electric field through a rectangle when the field is defined by a vector function of one, two, or three coordinates. Indicator 19.a.3 State and apply the relationship between flux and lines of force. Benchmark 19.b Students should understand Gauss’ Law. Indicator 19.a.1 State the law in integral form, and apply it qualitatively to relate flux and electric charge for a specified surface. Indicator 19.a.1 State the in differential form, and apply it qualitatively to relate electric field and volume charge density for a specified surface. Indicator 19.a.1 Apply the integral form of the law, along with symmetry arguments, to determine the electric field for a planar, spherical, or cylindrically symmetric charge distribution including some non-uniform charge distributions. Indicator 19.a.1 Apply the integral form to determine the charge density or total charge on a surface in terms of the electric field near the surface. Standard 20: Fields and potentials of other charge distributions Benchmark 20.a Students should be able to use the principle of superposition Indicator 20.a.1 Calculate by integration the electric field of a straight, uniformly charged wire. Electricity and Magnetism Course Electrostatics 19 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 20.a.2 Calculate by integration the electric field and potential on the axis of a thin ring of charge, or at the center of a circular arc of charge. Indicator 20.a.3 Calculate by integration the electric potential on the axis of a uniformly charged disk. Benchmark 20.b Students should know the fields of highly symmetric charge distributions Indicator 20.b.1 Identify situations in which the direction of the electric field produced by a charge distribution can be deduced from symmetry considerations. Indicator 20.b.2 Describe qualitatively the patterns and variation with distance of the electric field of oppositely charged parallel plates. Indicator 20.b.3 Describe qualitatively the patterns and variation with distance of the electric field of a long, uniformly-charged wire, or thin cylindrical or spherical shell. Indicator 20.b.4 Use superposition to determine the fields of parallel charged planes, coaxial cylinders, or concentric spheres. Indicator 20.b.5 Derive expressions for electric potential as a function of position in the above cases. Conductors, capacitors, and dielectrics Standard 21: Electrostatics with conductors Benchmark 21.a Students should understand the nature of electric fields in and around conductors. Indicator 21.a.1 Explain the mechanics responsible for the absence of electric field inside a conductor, and know that all excess charge must reside on the surface of the conductor. Indicator 21.a.2 Explain why a conductor must be an equipotential, and apply this principle in analyzing what happens when conductors are connected by wires. Indicator 21.a.3 Show that all excess charge on a conductor must reside on its surface and that the field outside the conductor must be perpendicular to the surface. Electricity and Magnetism Course Conductors, capacitors, and dielectrics 20 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 21.a.4 Students should be able to describe and sketch a graph of the electric field and potential inside and outside a charged conducting sphere. Benchmark 21.b Students should understand induced charge and electrostatic shielding. Indicator 21.b.1 Describe the process of charging by induction. Indicator 21.b.2 Explain why a neutral conductor is attracted to a charged object. Indicator 21.b.3 Explain why there can be no electric field in a charge-free region completely surrounded by a single conductor, and recognize consequences of this result. Indicator 21.b.4 Explain why the electric field outside a closed conducting surface cannot depend on the precise location of charge in the space enclosed by the conductor, and identify consequences of this result. Standard 22: Capacitors Benchmark 22.a Students should understand the definition and function of capacitance. Indicator 22.a.1 Relate stored charge and voltage for a capacitor. Indicator 22.a.2 Relate voltage, charge, and stored energy for a capacitor. Indicator 22.a.3 Recognize situations in which energy stored in a capacitor is converted to other forms. Benchmark 22.b Students should understand the physics of the parallel-plate capacitor. Indicator 22.b.1 Describe the electric field inside the capacitor, and relate the strength of this field to the potential difference between the plates and the plate separation. Indicator 22.b.2 Relate the electric field to the density of the charge on the plates. Indicator 22.b.3 Derive an expression for the capacitance of a parallel-plate capacitor. Indicator 22.b.4 Determine how changes in dimension will affect the value of the capacitance. Electricity and Magnetism Course Conductors, capacitors, and dielectrics 21 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 22.b.5 Derive and apply expressions for the energy stored in a parallel-plate capacitor and for the energy density in the field between the plates. Indicator 22.b.6 Analyze situations in which capacitor plates are moved apart or moved closer together, or in which a conducting slab is inserted between capacitor plates, or plates are not parallel, either with a battery connected between the plates or with the charge on the plates held fixed. Benchmark 22.b Students should understand cylindrical and spherical capacitors. Indicator 22.b.1 Calculate and describe the electric field inside both cylindrical and spherical capacitors. Indicator 22.b.2 Derive an expression for the capacitance of both cylindrical and spherical capacitors. Benchmark 22.c Students should understand the behavior of dielectrics. Indicator 22.c.1 Describe how the insertion of a dielectric between the plates of a charged parallel-plate capacitor affects its capacitance and the field strength and voltage between the plates. Indicator 22.c.2 Analyze situations in which a dielectric slab is inserted between the plates of a capacitor. Indicator 22.c.3 Analyze situations in which the dielectric has non-uniform geometries or dielectrics only partially fill the space between the sides of the capacitor. Electric circuits Standard 23: Current, resistance, and power Benchmark 23.a Students should understand the definition of electric current, so they can relate the magnitude and direction of the current to the rate of flow of positive and negative charge. Students should also understand conductivity, resistivity and resistance. Indicator 23.a.1 Relate current and voltage for a resistor. Electricity and Magnetism Course Electric circuits 22 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 23.a.2 Write the relationship between electric field strength and current density in a conductor, and describe, in terms of the drift velocity of electrons, why such a relationship is plausible. Indicator 23.a.3 Use the relationship between electric field strength and current density in combination with integration to calculate electronic properties of materials with non-uniform but symmetric resistivity. Indicator 23.a.4 Describe how the resistance of a resistor depends upon its length and cross-sectional area, and apply this result in comparing current flow in resistors of different material or different geometry. Indicator 23.a.5 Derive an expression for the resistance of a resistor of uniform or non-uniform crosssection in terms of its dimensions and the resistivity of the material from which it is constructed. Indicator 23.a.6 Derive expressions that relate the current, voltage and resistance to the rate at which heat is produced when current passes through a resistor. Indicator 23.a.7 Apply the relationships for the rate of heat production in a resistor. Standard 24: Steady-state direct current circuits with batteries and resistors only Benchmark 24.a Students should understand the behavior of series and parallel combinations of resistors. Indicator 24.a.1 Identify on a circuit diagram whether resistors are in series or in parallel. Indicator 24.a.2 Determine the ratio of the voltages across resistors connected in series or the ratio of the currents through resistors connected in parallel. Indicator 24.a.3 Calculate the equivalent resistance of a network of resistors that can be broken down into series and parallel combinations. Indicator 24.a.4 Calculate the voltage, current, and power dissipation for any resistor in such a network of resistors connected to a single power supply. Indicator 24.a.5 Design a simple series-parallel circuit that produces a given current through and potential difference across one specified component, and draw a diagram for the circuit using conventional symbols. Electricity and Magnetism Course Electric circuits 23 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Benchmark 24.b Students should understand the properties of ideal and real batteries. Indicator 24.b.1 Calculate the terminal voltage of a battery of specified emf and internal resistance from which a known current is flowing. Indicator 24.b.2 Calculate the rate at which a battery is supplying energy to a circuit or is being charged up by a circuit. Benchmark 24.c Students should be able to apply Ohm’s law and Kirchhoff’s rules to directcurrent circuits. Indicator 24.c.1 Determine a single unknown current, voltage, or resistance. Indicator 24.c.2 Set up and solve simultaneous equations to determine unknown currents. Benchmark 24.d Students should understand the properties of voltmeters and ammeters. Indicator 24.d.1 State whether the resistance of each is high or low. Indicator 24.d.2 Identify or show correct methods of connecting meters into circuits in order to measure voltage or current. Indicator 24.d.3 Assess qualitatively the effect of finite meter resistance on a circuit into which these meters are connected. Standard 25: Capacitors in circuits Benchmark 25.a Students should understand the t = 0 and steady-state behavior of capacitors connected in series or in parallel. Indicator 25.a.1 Calculate the equivalent capacitance of a series or parallel combination. Indicator 25.a.2 Describe how stored charge is divided between capacitors connected in parallel. Indicator 25.a.3 Determine the ratio of voltages for capacitors connected in series. Electricity and Magnetism Course Electric circuits 24 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 25.a.4 Calculate the voltage or stored charge, under steady-state conditions, for a capacitor connected to a circuit consisting of a battery and resistors. Benchmark 25.b Students should understand the discharging or charging of a capacitor through a resistor. Indicator 25.b.1 Calculate and interpret the time constant of the circuit. Indicator 25.b.2 Sketch or identify graphs of stored charge or voltage for the capacitor, or of current or voltage for the resistor, and indicate on the graph the significance of the time constant. Indicator 25.b.3 Write expressions to describe the time dependence of the stored charge or voltage for the capacitor, or of the current or voltage for the resistor. This includes the development and solutions of ordinary differential equations that can describe a variety of RC circuits. Indicator 25.b.4 Analyze the behavior of circuits containing several capacitors and resistors, including analyzing or sketching graphs that correctly indicate how voltages and currents vary with time. Magnetic fields Standard 26: Magnetic forces on moving charges in magnetic fields Benchmark 26.a Students should understand the force experienced by a charged particle in a magnetic field. Indicator 26.a.1 Calculate the magnitude and direction of the force in terms of q, v, and B, and explain why the magnetic force can perform no work. Indicator 26.a.2 Deduce the direction of a magnetic field from information about the forces experienced by charged particles moving through that field. Indicator 26.a.3 Describe the paths of charged particles moving in uniform magnetic fields. Indicator 26.a.4 Derive and apply the formula for the radius of the circular path of a charge that moves perpendicular to a uniform magnetic field. In addition expand this understanding to charged particles moving in helical paths. Electricity and Magnetism Course Magnetic fields 25 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 26.a.5 Describe under what conditions particles will move with constant velocity through crossed electric and magnetic fields. Standard 27: Magnetic forces on current-carrying wires in magnetic fields Benchmark 27.a Students should understand the force exerted on a current-carrying wire in a magnetic field. Indicator 27.a.1 Calculate the magnitude and direction of the force on a straight segment of currentcarrying wire in a uniform magnetic field. Indicator 27.a.2 Indicate the direction of magnetic forces on a current-carrying loop of wire in a magnetic field, and determine how the loop will tend to rotate as a consequence of these forces. Indicator 27.a.3 Calculate the magnitude and direction of the torque experienced by a rectangular loop of wire carrying a current in a magnetic field. Indicator 27.a.4 Apply concepts of torque on wires due to magnetic forces to situations that result in simple harmonic motion including the use of small angle approximations. Standard 28: Magnetic fields of long current-carrying wires Benchmark 28.a Students should understand the magnetic field produced by a long straight current-carrying wire. Indicator 28.a.1 Calculate the magnitude and direction of the field at a point in the vicinity of such a wire. Indicator 28.a.2 Use superposition to determine the magnetic field produced by two long wires. Indicator 28.a.3 Calculate the force of attraction or repulsion between two long current-carrying wires. Standard 29: Biot-Savart law and Ampere’s Law Benchmark 29.a Students should understand the Biot-Savart Law. Indicator 28.a.1 Deduce the magnitude and direction of the contribution to the magnetic field made by a short straight segment of current-carrying wire. Electricity and Magnetism Course Magnetic fields 26 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 29.a.2 Derive and apply the expression for the magnitude of B on the axis of a circular loop of current. Indicator 29.a.3 Derive and apply the expression for the magnitude and direction of B on the center axis of a solid disk, solenoid, and Helmholtz coils. Benchmark 29.b Students should understand the statement and application of Ampere’s Law. Indicator 29.b.1 State the law precisely in both integral and differential form. Indicator 29.b.2 Use Ampere’s law, plus symmetry arguments and the right-hand rule, to relate magnetic field strength to current for planar or cylindrical symmetries. Indicator 29.b.3 Students should be able to apply the superposition principle so they can determine the magnetic field produced by combinations of the configurations listed above. Electromagnetism Standard 30: Electromagnetic induction including Faraday’s law and Lenz’s law Benchmark 30.a Students should understand the concept of magnetic flux. Indicator 30.a.1 Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation. Indicator 30.a.2 Use integration to calculate the flux of a non-uniform magnetic field, whose magnitude is a function of one coordinate, through a rectangular loop perpendicular or otherwise to the field. Benchmark 30.b Students should understand Faraday’s law and Lenz’s law. Indicator 30.b.1 Recognize situations in which changing flux through a loop will cause an induced emf or current in the loop. Indicator 30.b.2 Calculate the magnitude and direction of the induced emf and current in a loop of wire or a conducting bar with either he magnitude of a related quantity such as magnetic field or area of the loop is changing at a constant rate or the magnitude of a related quantity such as magnetic field or area of the loop is a specified non-linear function of time. Electricity and Magnetism Course Electromagnetism 27 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 30.b.3 Analyze the forces that act on induced currents so they can determine the mechanical consequences of those forces. Indicator 30.b.4 Apply the concepts of magnetic inductance to analyze simple transformers Standard 31: Inductance including LR and LC circuits Benchmark 31.a Students should understand the concept of inductance. Indicator 31.a.1 Calculate the magnitude and sense of the emf in an inductor through which a specified changing current is flowing. Indicator 31.a.2 Derive and apply the expression for the self-inductance of a long solenoid. Benchmark 31.b Students should understand the transient and steady state behavior of DC circuits containing resistors and inductors. Indicator 31.b.1 Apply Kirchhoff’s rules to a simple LR series circuit to obtain and solve a differential equation for the current as a function of time. Indicator 31.b.2 Calculate the initial transient currents and final steady state currents through any part of a simple series and parallel circuit containing an inductor and one or more resistors. Indicator 31.b.3 Sketch graphs of the current through or voltage across the resistors or inductor in a simple series and parallel circuit. Indicator 31.b.4 Calculate the rate of change of current in the inductor as a function of time. Indicator 31.b.5 Calculate the energy stored in an inductor that has a steady current flowing through it. Benchmark 31.c Students should understand the oscillating behavior of circuits containing capacitors and inductors. Indicator 31.c.1 Apply Kirchhoff’s rules to a simple LC series circuit to obtain and solve a differential equation for the current as a function of time including the calculation of oscillating frequency and maximum current. Electricity and Magnetism Course Electromagnetism 28 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 31.c.2 Describe the electrical-mechanical analogy for LC, LRC and driven LRC circuits. Benchmark 31.d Students should be familiar with Maxwell’s equations so they can associate each equation with its implications. Indicator 31.d.1 Define both the integral and differential form of Maxwell’s equations including Maxwell’s expansion of Ampere’s Law using displacement current. Indicator 31.d.2 Starting from the differential form of Maxwell’s equations show that they combine to form a differential equation that predicts the wave nature of light including the speed of light in a vacuum. Common Experimental Standards Laboratory investigations Standard 32: Design experiments Benchmark 32.a Students should understand the process of designing experiments. Indicator 32.a.1 Describe the purpose of an experiment or a problem to be investigated. Indicator 32.a.2 Identify equipment needed and describe how it is to be used. Indicator 32.a.3 Draw a diagram or provide a description of an experimental setup. Indicator 32.a.4 Describe procedures to be used, including controls and measurements to be taken. Standard 33: Observe and measure real phenomena Benchmark 33.a Students should be able to make relevant observations, and be able to take measurements with a variety of instruments. Indicator 33.a.1 Record and communicate to peers observations made using a variety of instruments. Standard 34: Analyze data Common Experimental Standards Laboratory investigations 29 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Benchmark 34.a Students should understand how to analyze data. Indicator 34.a.1 Display data in graphical or tabular form. Indicator 34.a.2 Fit lines and curves to data points in graphs, which may include using appropriate linearization techniques. Indicator 34.a.3 Perform calculations with data. Indicator 34.a.4 Make extrapolations and interpolations from data. Standard 35: Analyze data Benchmark 35.a Students should understand measurement and experimental error. Indicator 35.a.1 Identify sources of error and how they propagate including the formal propagation of error through calculations using one of several methods. Indicator 35.a.2 Estimate magnitude and direction of errors. Indicator 35.a.3 Determine significant digits. Indicator 35.a.4 Identify ways to reduce error. Standard 36: Communicate results Benchmark 36.a Students should understand how to summarize and communicate results. Indicator 36.a.1 Draw inferences and conclusions from experimental data. Indicator 36.a.2 Suggest ways to improve experiment. Indicator 36.a.3 Propose questions for further study. Common Experimental Standards Laboratory investigations 30 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Teacher Specific Supplemental Topics Waves and optics Standard 37: Traveling waves Benchmark 37.a Students should understand the description of traveling waves. Indicator 37.a.1 Sketch or identify graphs that represent traveling waves and determine the amplitude, wavelength, and frequency of a wave from such a graph. Indicator 37.a.2 Apply the relation among wavelength, frequency, and velocity for a wave. Indicator 37.a.3 Understand qualitatively the Doppler effect for sound in order to explain why there is a frequency shift in both the moving-source and moving-observer case. Indicator 37.a.4 Describe reflection of a wave from the fixed or free end of a string. Indicator 37.a.5 Describe qualitatively what factors determine the speed of waves on a string and the speed of sound. Standard 38: Wave propagation Benchmark 38.a Students should understand how waves propagate through a medium. Indicator 38.a.1 Qualitatively explain the difference between transverse and longitudinal waves. Indicator 38.a.2 Qualitatively describe why transverse waves can exhibit polarization and longitudinal cannot. Indicator 38.a.3 Using the inverse square relationship, calculate wave intensity at a give distance from a source of specified power. Indicator 38.a.4 Using Maxwell’s equations as a basis, justify how the propagation of electromagnetic waves does not require a medium. Indicator 38.a.5 Relate a variation of index of refraction with frequency to a variation in refraction. Teacher Specific Supplemental Topics Waves and optics 31 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Standard 39: Standing waves Benchmark 39.a Students should understand the physics of standing waves. Indicator 39.a.1 Sketch possible standing wave modes for a stretched string that is fixed at both ends, and determine the amplitude, wavelength, and frequency of such standing waves. Indicator 39.a.2 Describe possible standing sound waves in a pipe that has either open or closed ends, and determine the wavelength and frequency of such standing waves. Indicator 39.a.3 Apply the principle of superposition to traveling waves moving in opposite directions to describe how a standing wave is formed. Standard 40: Interference and diffraction Benchmark 40.a Students should understand the interference and diffraction of waves. Indicator 40.a.1 Apply the principles of interference to coherent sources in order to describe the conditions under which the waves reaching an observation point from two or more sources will all interfere constructively, or under which the waves from two sources will interfere destructively. Indicator 40.a.2 Apply the principles of interference to coherent sources in order to determine locations of interference maxima or minima for two sources or determine the frequencies or wavelengths that can lead to constructive or destructive interference at a certain point. Indicator 40.a.3 Apply the principles of interference to coherent sources in order to relate the amplitude produced by two or more sources that interfere constructively to the amplitude and intensity produced by a single source. Indicator 40.a.4 Sketch or identify the intensity pattern that results when monochromatic waves pass through a single slit and fall on a distant screen, and describe how this pattern will change if the slit width or the wavelength of the waves is changed. Indicator 40.a.5 Calculate, for a single-slit pattern, the angles or the positions on a distant screen where the intensity is zero. Indicator 40.a.6 Sketch or identify the intensity pattern that results when monochromatic waves pass through a double slit, and identify which features of the pattern result from single-slit diffraction and which from two-slit interference. Teacher Specific Supplemental Topics Waves and optics 32 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 40.a.7 Calculate, for a two-slit interference pattern, the angles or the positions on a distant screen at which intensity maxima or minima occur. Indicator 40.a.8 Describe or identify the interference pattern formed by a diffraction grating, calculate the location of intensity maxima, and explain qualitatively why a multiple-slit grating is better than a two-slit grating for making accurate determinations of wavelength. Standard 41: Geometric optics Benchmark 41.a Students should understand the principles of reflection and refraction. Indicator 41.a.1 Determine how the speed and wavelength of light change when light passes from one medium into another. Indicator 41.a.2 Show on a diagram the directions of reflected and refracted rays. Indicator 41.a.3 Use Snell’s Law to relate the directions of the incident ray and the refracted ray, and the indices of refraction of the media. Indicator 41.a.4 Identify conditions under which total internal reflection will occur. Introduction to modern physics Standard 42: Wave particle duality Benchmark 42.a Students should understand the concept of de Broglie wavelength. Indicator 42.a.1 Calculate the wavelength of a particle as a function of its momentum. Indicator 42.a.2 Describe the Davisson-Germer experiment, and explain how it provides evidence for the wave nature of electrons. Standard 43: Special relativity Benchmark 43.a Students should understand the deep implications of the two postulates of special relativity. Indicator 42.a.1 Apply the concepts of special relativity to describe issues surrounding simultaneity. Teacher Specific Supplemental Topics Introduction to modern physics 33 Thomas Jefferson High School for Science and Technology Program of Study – AP Physics C: Mechanics and Electricity & Magnetism Last Revised July 2016 Indicator 42.a.2 Quantitatively apply the concepts of time dilation and length contraction. Standard 44: Mass energy equivalence Benchmark 44.a Students should understand the relationship between mass and energy Indicator 44.a.1 Qualitatively relate the energy released in nuclear processes to the change in mass. Indicator 44.a.2 Apply the relationship in analyzing nuclear processes. Indicator 44.a.3 Quantitatively describe relativistic energy and momentum as particles approach the speed of light and relate these expressions to their classical counterparts through the application of the Principle of Complementanty. Teacher Specific Supplemental Topics Introduction to modern physics 34