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Electric Charge, Electric Forces, and Electric Fields Young and Freedman Chapter 21 Elemental Particles and Charge • Atoms form the building blocks of matter • Atoms are comprised of elemental particles: protons, neutrons and electrons – Charge of proton is positive – Charge of electron is negative – Charge of neutron is …. well … neutral • Charge is a scalar quantity • Proton charge = - electron charge (equal size, but opposite sign) 1) Charge is conserved: The total charge in a closed system remains constant 2) Charge is measured in integer units of electron charge Conceptual Aside If a nucleus was the size of a tennis ball, where would the atomic cloud be? Conceptual Aside • If we were to scale up the nucleus to the size of a tennis ball (say a couple of inches or 5cm), where would the electron cloud be? • Nuclear size in Fermis or 10-15 m • Atomic size in Angstroms or 10-10 m • Five orders of magnitude difference! i.e atom is approximately 100,000 times bigger than the nucleus • 100,000 x 5 cm = 500,000 cm = 5 km ~ 3 miles So, if the nucleus was here in this room, the atomic cloud would stretch almost to the UT Medical Center .... and there is nothing in between! Charge Balance • There are approximately 1028 electrons in your body • Why aren’t we all highly charged???? • Approximately equal numbers of protons and electrons • Remember, charge of proton and electron has same magnitude, but opposite sign • So, on balance we are more or less electrically neutral • If something has a non-zero charge, it has an imbalance in the number of positive and negative charges Coulomb’s Law • • • • Coulomb discovered that the forces that two point charges (we call them q1 and q2) exert on each other is proportional to the product q1q2 It is also inversely proportional to the square of the distance between them (r2) Remember, the force is attractive if the charges have different signs and repulsive if the signs are the same. Force is a vector! The magnitude of the force can be written as: q1q2 F=k 2 r F (force) is in Newtons (N) q (charge) is in Coulombs (C) r (distance) is in Meters (m) k is a constant and must have units of N.m2 / C2 Coulomb’s Law - vector form • • Force is a vector quantity Direction depends on whether charges are: ‒ [1] like (both positive, or both negative), or – [2] one positive and one negative [1] like charges r F= [2] unlike charges 1 q1q2 ˆ r 2 4 "#0 r ! 2 N • m k = 8.988 "10 9 C2 Principle of Superposition • • Simply put it means that we can sum the forces due to individual charges but .... remember we’re dealing with vectors so this is a vector sum To determine the force on charge q from a number of charges q1, q2, q3, ..... r F= 1 qq1 1 qq2 1 qq3 ˆ ˆ ˆ 2 r1 + 2 r2 + 2 r3 + ....... 4 "#0 r1 4 "#0 r2 4 "#0 r3 r 1 qqi ˆ F =$ r 2 i i 4 "# 0 ri Coulomb’s Law/Superposition: Example Example 21.4 from the book First look at the force exerted on Q by q1 Take the components in x and y 1 = 9.0 $10 9 N % m 2 /C 2 4 "# 0 Note that x is positive and y negative in this case. IMPORTANT these are vectors y q1=2.0µC + ! 0.5m 0.3m Q=4.0µC 0.4m 0.3m + 0.5m q2=2.0µC + x Coulomb’s Law/Superposition: Example Example 21.4 from the book First look at the force exerted on Q by q1 (4.0 "10 #6 )(2.0 "10 #6 ) F1 = (9.0 "10 ) N 2 (0.5) = 0.29N 9 1 = 9.0 $10 9 N % m 2 /C 2 4 "# 0 y q1=2.0µC Take the components in x and y 0.40 F1x = F1 cos " = 0.29 N = 0.23N ! 0.50 0.30 F1y = F1 sin " = 0.29 N = #0.17N 0.50 + 0.5m 0.3m Q=4.0µC 0.4m 0.3m + 0.5m q2=2.0µC + x Coulomb’s Law/Superposition: Example Example 21.4 from the book Now look at the force exerted on Q by q2 Use logic to work out from previous calculations what happens to the x and y components F2x is the same magnitude and direction as F1x F2y is the same magnitude but opposite ! direction to F1y Hence, Fxtot= 2 x F1x and Fytot = 0 The total force is in the x-direction and has a magnitude of 0.46 N 1 = 9.0 $10 9 N % m 2 /C 2 4 "# 0 The Electric Field • Rearranging Coulomb’s law for the force exerted on a charge q by a collection of charges q1, q2, ... qi r 1 qi ˆ F = q$ r 2 i i 4 "# 0 ri r r F = qE We can write the vector equation for the electric field of a point charge as: ! r E= 1 q ˆ r 2 4 "# 0 r Concept: The Test Charge • A test charge is a conceptual tool used to calculate the properties of electric fields • We assume that the value of the charge (q0) is vanishingly small, hence it doesn’t affect the field • However q0 is finite as it does feel the effect of the field • We imagine placing a test charge q0 at a point P (see examples in text) • We consider the force F0 on this test charge due to the field from the other charge(s) • This leads to the definition of therelectric field r F0 E = lim q0 "0 q0 Electric Field - Test Charge r E = lim q"0 r F0 q0 Definition of electric field A point charge creates an electric field at all points in space (note these are E vectors not field lines, we’ll meet those soon) The Electric Field of Point Charges r E= 1 q ˆ r 4 "# 0 r 2 The Electric Field of a Point Charge We can calculate the Electric Field produced by a collection of charges: r 1 qi ˆ E =$ r 2 i i 4 "# 0 ri r 1 qi ˆ F = q0 $ r 2 i 4 "# r 0 i i Compare with: Coulomb’s Law (superposition) Electric Field - Continuous Distribution of Charges If the charges are not point charges, but are distributed continuously we r can replace qi by the infinitesimal charge dq(r ) and write: r 1 dq(r ) E= $ rˆ 2 r all!4 "# 0 A useful concept is that of charge density per unit volume) given by: r dq(r) = " (r )dv ! " (rˆ ) (the amount of charge Where dv is an infinitesimal element of volume r 1 % (r ) ˆ E= $ r dv 2 all 4 "# 0 r ! Definitions of electric field directions • The electric field of a point charge always points away from a positive charge, but toward a negative charge • The electric field varies from point to point, both in magnitude and direction • Electric field is a vector field Example 21.10: Calculating Electric Field of a Ring of Charge • STEP 1 -Draw a sketch of the system • STEP 2 - Consider a tiny segment of the charged ring with charge dQ, as a point charge • STEP 3 - Calculate the field dE that results from dQ • STEP 4 - Integrate over all such segments r E= r " dE The magnitude of dE is needed to calculate the x and y components 1 dQ dE = 4 "# 0 x 2 + a 2 The x-component dEx=dE cos α; and cos α = x/r 1 dQ dE x = 4 "# 0 x 2 + a 2 ! ! 1 dE x = 4 "# 0 x 2 x +a xdQ 3 2 2 (x 2 + a ) 2 Integrate to find Ex Ex = $ 1 4 "# 0 xdQ 3 2 2 (x 2 + a ) x doesn’t vary as we integrate around the ring hence ! 1 Ex = 4 "# 0 Qx 3 2 2 (x 2 + a ) Symmetry means that the y-components cancel out with those diagonally opposite, hence: r E = E x iˆ = 1 4 "# 0 Qx 2 3 2 2 (x + a ) iˆ r E = E x iˆ = 1 4 "# 0 Qx 2 3 2 2 iˆ (x + a ) Note: at a large distance, such that x>>a, the denominator becomes x3 and the expression approximates to: r E= ! 1 Qˆ i 2 4 "# 0 x i.e. it looks like a point charge, as would be expected ! Electric Field Lines • Electric Field Lines are used to help visualize an electric field • The local direction of the field line is the direction of the E (or F) vector at that point • In general, the direction of the electric field is a tangent to the electric field line at that point. Field Lines • The density of the lines represents the magnitude of the electric field • At any point the field has a unique vector, hence only one field line can be present at any point in space • This leads to the fact that field lines do not intersect • In general, the field line does not represent the direction of motion of a charged particle at that point Short Note on Conductors and Insulators • A CONDUCTOR is a material in which electric charges are free to move around • An INSULATOR is a material in which electric charges are immobile • Both conductors and insulators can be charged or uncharged (remember this refers to the NET charge) Electric Field Inside A Conductor • • Charges can move freely inside a conductor Assume there is an electric field inside the conductor • Since the charges are free to move, they will be pushed toward the surface of the conductor - + + • • - - - ++ The field of these charges will tend to cancel the original field They will continue to move until the field is zero The Electric Field inside a Conductor is EXACTLY ZERO End of Chapter 21 You are expected to: • Know what the following are: – charge, electrons, protons, atomic number, conductors, insulators, electrostatic force, superposition, and electric field • • • • Understand Coulomb’s Law, unit vectors and be able to express the direction of an electric field Calculate the force on a point charge due to another point charge Calculate the the electric field from a point charge due to another point charge and from a simple collection of point charges Understand the notion of Electric Field Lines and be able to roughly sketch the field lines for simple collections of charges Chapter 21 Summary Chapter 21 Summary cont.