Electricity and Magnetism
Unit Test Review
General Guidelines
• Reading the book or notes (even these below) is passive studying. It doesn’t help
much.
• Active studying means putting all of your faculties to work so write things down, test
yourself, get involved in your studying!
• Study your unit quizzes by retaking them. Rewrite the questions if you need to.
• Make flash cards out of the information below. Put a question on the back of each
one and test yourself.
Electrostatics Review
• Electrical Force – Coulomb’s Law
F =
kq1 q2
R2
(1)
where k is Coulomb’s constant (k = 8.99 · 109 N m2 / C2 ), q1 and q2 are charges and R
is the distance between the charges.
– The force on particle is towards or away from the charge center of the other
particle (don’t forget equal and opposite).
– Doubling either charge doubles the force.
– Doubling the distance cuts the force to one quarter.
• The Three Rules of Electrostatics
– There are only two kinds of charge.
– Two objects charged alike repel each other
– Two objects charged oppositely attract each other.
• Fields
– A field is a representation of a function throughout space; a function of x, y and
z.
– A force field represents a force that would affect an appropriate object placed in
the field.
– A scalar field just has a single value at every point.
– A vector field has a value and a direction at every point. All force fields are
vector fields!
c 2008 Steve Goldhaber – Physics Honors
• Force due to an electric field:
~
F~ = q E
(2)
– The electric field vector points in the direction that it would push a positive
charge.
– The electric field exists at all points in space.
– The electric field at some point in space (x, y, z) due to many charges is just
the vector sum of the electric field at that point due to each charge (principle of
superposition).
• The electric field around a point charge points away from the charge (for a positive
charge) and towards the charge (for a negative charge).
• Electric field lines point in the direction of the electric field at every point. They are
drawn closer together when the field is stronger.
Figure 1: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html (2/28/2007)
Terms:
Term
Unit
Common Symbol
Electric Potential Difference (voltage) Volt (V)
V
Electric Current
Ampere (Amp or A) A
Resistance
Ohm (Ω)
R
Capacitance
Farad (F)
C
Laws governing direct-current electric circuits
• Ohm’s Law: V = IR
Ohm’s law is the basic relationship which shows the relationship between the current
through a resistor (or other type of resistive element such as a light bulb) and the
voltage drop across it.
c 2008 Steve Goldhaber – Physics Honors
• Kirchoff ’s Circuit Junction Law: The sum of all current entering a junction must
be zero. This can also be stated as: The amount of current entering a junction equals
the total amount of current leaving the junction. What goes in must come out. To
use this law, observe these rules:
– Draw the direction for the current in each section of the circuit. For complicated
circuits, this may not be obvious. In that case, pick a direction and draw it.
– Add up all the currents entering a junction and set it equal to the sum of all the
currents leaving a junction. This gives you one of the equations you will need to
analyze a circuit.
– If at the end of your analysis, any of your currents turn out to be negative, that
just means you got the direction of that current backwards. It does not mean
you made a mistake.
• Kirchoff ’s Voltage Loop Law: The sum of voltage drops around any closed circuit
loop must be zero. To use this law, observe these rules:
– You must pick a direction in which to travel the loop (clockwise or counterclockwise). It does not matter which but you must be careful to keep the direction
constant.
– Sum the voltage of each and every element in the loop and set the sum equal to
zero.
– When traversing a voltage source (e.g., a battery) from negative to positive, you
add in the voltage as a positive contribution to the sum (i.e., use a positive sign).
– When traversing a voltage source (e.g., a battery) from positive to negative, you
subtract the voltage from the sum (i.e., use a negative sign).
– When traversing a resistive element in the direction of the current, subtract the
voltage from the sum.
– When traversing a resistive element against the direction of the current, add the
voltage to the sum.
• Power generated in a circuit is the current times the voltage (P = IV ).
• Power dissipated (i.e., turned into heat) by a resistive element of a circuit is P =
I 2 R = V 2 /R where I is the current through the resistor and V is the voltage drop
across the resistor.
• A voltmeter measures the potential difference (voltage) between two points in a circuit.
The voltmeter should not disturb a circuit because it has very high internal resistance.
• An ammeter measures the current in a portion of the a circuit. The ammeter must be
made part of the circuit (i.e., the current to be measured must pass through the ammeter). A good ammeter should not disturb the circuit because its internal resistance
is very low (as close to zero as possible).
• Electrical power is mostly created by generators. They turn mechanical energy into
electrical energy.
c 2008 Steve Goldhaber – Physics Honors
• Electric motors turn electrical energy into mechanical energy. The chart below should
illustrate the differences:
device
Alternating Current (AC) Direct Current (DC)
Motor
Electrical Energy → Mechan- Electrical Energy → Mechaniical Energy. No commutator cal Energy. Has a commutator
(requires AC input)
(requires DC input)
Generator Mechanical Energy → Electri- Mechanical Energy → Electrical Energy. No commutator cal Energy. Has a commutator
(produces AC output)
(produces DC output)
When combining two resistors in series, the equivalent resistance is just the sum of the
two resistors:
Req = R1 + R2
To combine resistors in parallel, the inverse of the equivalent resistance is equal to the sum
of the inverses of the parallel resistors.
1
1
1
=
+
Req
R1 R2
Magnet Basics
• A magnet is an object which creates a magnetic field.
• All known magnets have both a North and a South pole.
• A magnet can attract or repel another magnet.
c 2008 Steve Goldhaber – Physics Honors, Figures courtesy of PIRA
Magnetic Fields
•
•
•
•
•
Magnetic field lines point from North to South.
Magnetic field lines are drawn closer together to represent a stronger field.
Magnetic field lines are not vectors.
The magnetic field is a vector field.
~
The symbol for the magnetic field is B.
Magnetic Forces
• The force on a moving charge in a magnetic field is:
~
F~ = q~v × B
(3)
~ = |q||~v ||B|
~ sin(θ)
F = |F~ | = |q|v⊥ |B|
(4)
• The magnitude of the force is:
• The direction of the force is found using the right hand rule.
1. Point your fingers in the direction of the velocity of the charged particle.
2. Curl your fingers in the direction of the magnetic field.
3. Your thumb points in the force direction.
• If the moving charge is negative, the direction of the Lorentz force will be in the
~
opposite direction from ~v × B.
• An important property of the right hand rule is that the magnetic Lorentz force will
always be perpendicular to both the charged particle’s velocity and to the magnetic
field in which it is traveling.
• Because of the previous property, if the particle’s initial velocity is perpendicular to
the magnetic field, the particle will travel in a circle. To find the radius of the circle,
compare the Lorentz force (equation 3) to the required centripital force (Fc = mac =
mv 2 /r).
• If the particle’s path is not perpendicular to the magnetic field (but is also not parallel
to it), the particle will follow a helical path. The component of the velocity perpendicular to the field causes circular motion (with the same radius calculation as shown
above) while the parallel component is unchanged.
Magnetic fields and conducting wires:
• The Lorentz force (equation 3) can be used to find the force on a current-carrying
wire.
F = I`B sin θ
(5)
where I is the current and ` is the length of the wire.
• To find the direction of force on a current-carrying wire, use the right hand rule with
the direction of the current playing the role of the velocity vector in the Lorentz force
equation.
c 2008 Steve Goldhaber – Physics Honors
Magnetic field due to an electric current
• When a wire is carrying a current, a magnetic field is created which circles the wire.
• If you point your thumb along the direction of current, your fingers will naturally curl
in the direction of the magnetic field
• The strength of the magnetic field near a current-carrying wire is:
B=
µ0 I
2πR
(6)
where µ0 = 4π · 10−7 N/ A2 , I is the current in the wire, and R is the distance from
the center of the wire.
• Because a wire creates a magnetic field and a parallel wire with a current contains
electric charge moving perpendicular to the first wire’s magnetic field, we can use
equations 5 and 6 to find the force exerted by each wire on the other. Let I1 be the
current in one wire and I2 be the current in the other wire which is parallel and a
distance, R, from the first wire. For a section of length `, we have:
F = I1 `B = I1 `(
µ0 I1 I2 `
µ0 I2
)=
2πR
2πR
(7)
• When you wrap a wire into a coil, a solenoid is created. A current in a solenoid creates
a magnetic field which is directed through the coil and is fairly uniform on the inside
and fairly weak on the outside.
• The magnetic field inside the coil is:
B = µ0 In
(8)
where I is the current in the solenoid and n is the number of turns per unit length
(i.e., turns / meter).
Electromagnetic Induction
• Electromagnetic induction is an electromotive force caused by a changing magnetic
field.
• Another statement of the property of electromagnetic induction is that a changing
magnetic field creates an electric field.
• A steady magnetic field will not create an electric field, nor will it create a current in
a wire.
• The changing magnetic field can be created by changing the strength of a magnetic
field or by moving a wire near a nonuniform magnetic field.
c 2008 Steve Goldhaber – Physics Honors
Magnetic Flux
• The magnetic flux through a surface is a way of adding up how much magnetic field
is ‘crossing’ the surface.
• The complete definition of magnetic flux is:
ZZ
~
~
ΦB = B
· dA
(9)
however, we will usually try to find conditions where the magnetic field is constant so
that we can simplify equation 9 to:
ΦB = BA cos(θ)
(10)
where θ is the angle between the magnetic field and a line perpendicular to the surface
whose area is A. For equation 10 to hold, the magnetic field must be a constant
everywhere on the surface and the angle between the surface and the magnetic field
must also be constant.
Faraday’s Law
• Faraday’s Law shows the relationship between a changing magnetic flux through a
loop or loops of wire and the electromagnetic force generated in the wire.
• Faraday’s Law is usually written as:
E = −N
dΦB
dt
(11)
where ΦB is computed using equation 9 or 10 and N is the number of loops.
Lenz’s Law
• The induced magnetic field always opposes the change in flux that caused it.
• Lenz’s Law is used to figure out the direction of the current induced by a changing
magnetic field.
c 2008 Steve Goldhaber – Physics Honors
Applications of Electromagnetic Induction
• An electric generator works by turning a coil in a magnetic field. This creates an
alternating current within the coil which can then be used as a source of electric
power.
• A transformer can be used to change the voltage of an alternating-current system. By
using a different number of turns of a coil in the supply side (primary) and the demand
side (secondary), the voltage can be changed. Since the power use does not change,
the current also changes. The relationships are:
Np
Vs
Ns
Ns
Ip =
Is
Np
Vp =
• A normal car brake turns the car’s kinetic energy into heat energy. An alternate form,
popular on hybrid cars, uses Lenz’s law to turn the car’s kinetic energy into electrical
energy which is then fed back to the car’s energy storage system (either the battery
or a capacitor).
• A microphone uses magnetic induction to convert mechanical waves (sound) into vibrations of a coil near a magnet which induces a current matching the frequencies of
the incident sound wave.
The Four principles of Electromagnetism
• An electric current in a conductor produces magnetic lines of force that circle the
conductor.
• When a conductor moves across externally set-up magnetic lines of force, a current is
induced in the conductor.
• A changing electric field in space produces a magnetic field.
• A changing magnetic field in space produces a electric field.
An electromagnetic wave is
• one of the predictions of Maxwell’s theory.
• a series of fluctuating, interlocked electric and magnetic fields that propagate through
space.
• a transverse wave with the displacement of the electric field separated 90◦ from the
displacement of the magnetic field.
• a phenomenon that comes in an infinite variety of frequencies (a spectrum) of which
visible light is a very small part.
Maxwell and the Electromagnetic wave
• Maxwell calculated what he thought the speed of the EM wave should be using wave
notions of density (magnetic field) and stiffness (electric field).
• He found that the strength of the fields did not affect the speed.
• He found that the frequency of the wave did not affect the speed.
• His calculated speed matches the speed of light!
c 2008 Steve Goldhaber – Physics Honors