AP Physics C E&M Objectives
AP Physics C E&M Standards
E. Electromagnetism
............................................................16%
1. Electromagnetic induction (including Faraday's
law and Lenz's law)
2. Inductance (including LR and LC circuits) *
3. Maxwell's equations *
Sources of Magnetic Fields
(continued)
Essential Question:
Can a magnetic field be used to
create a force on an object? 
Solenoids

Explain Ampere's law. Relevance:
Like Gauss's Law, Ampere's Law can be used in deriving other useful equations
such as finding the magnetic field generated by a solenoid. These devices are
used in numerous applications from door bells to the starter on cars.
∮B • ds
=
m_{o }I
 Calculate the B Field inside the object shown
below: Keep in mind that both the solenoid and
torroid are like stacks of current carrying rings. The magnetic field inside a
current carrying ring is strong, but outside the ring is very weak. In the
derivations of the equations for solenoids, The magnetic field outside the
coils (or rings) is assumed to be zero.

torroid: 
B = 
m_{o }
N_{ }I

2pr 

solenoid: 
B = 
m_{o }
N I

L 

current carrying wire: 
B = 
m_{o }
I r

2pR^{2} 
Homefun Prob 23, 25 p.895
Essential Question: How can a magnetic field be used to
generate electricity? 
Magnetic Flux
 Mathematically define magnetic fluxroughly
speaking, the amount of magnetic field passing through a given surface.
F_{B} = ò
B•dA
Note:
both B and A are vectors. The direction of the area vector is perpendicular
to the plane of the area. Since the equation is the integral of a dot
product of two vectors, magnetic flux is
a scalar.
 Explain briefly how a transformer works and why it requires an AC input.
N_{out} / N_{in} = V_{out} / V_{in}
 Calculate the magnetic flux through a rectangular loop of wire next
to a long thin current carrying wire in the same plane (p.850).
F_{B}= [ ( m_{o} I b) / 2p ] ln (1 + a/c)
 State Gauss's law for magnetism.
Why is the magnetic flux through a closed surface zero?
Because, unlike Efields, magnetic fields form closed loops so that a flux
line leaving the inside of a closed surface (giving it a positive value) will
eventually come back around and enter the surface giving it a negative value).
Hence, the two cancel each other out.
∮B•dA
= 0
Relevance:
Transformers which make use of magnetic flux to raise and lower voltages are
the basis of our electrical transmission system and the reason it uses AC
rather than DC power. 
Use a Tesla coil to light up a florescent tube from a distance.
Questions:
 Both a Tesla coil and a Van de Graaff Generator generate high
voltages. What is the difference between a Van de Graaff Generator and
Tesla coil?


Essential Question:
How would society be different if
Faraday's Law of induction did not exist ? 
Faraday's Law (Chap31 Serway)
 State and apply Faraday's Law of Induction. Describe
how magnetic flux can be altered with respect to time. Relevance:
Faraday's law describes how electricity can be generated by rotating a coil
in a magnetic field.
e =  dF_{B}
/ dt
Note: since both time and magnetic flux are
scalars the potential or voltage must also be a scalar, which, of course, it is
since it's a form of energy.
Homefun (formative summative
assessment): prob 1, 5 p.927
Formal Physics Investigation 
Title 
Measurement of the acceleration due to
gravity using a solenoid. 
Purpose 
To observe Faraday's law of induction by
dropping a magnet through a solenoid. 
Overview 
 Connect the terminals of the solenoid to the the LabPro voltage
probe.
 Place the tip of the magnet at the entrance to the solenoid
 Drop the magnet completely through the solenoid and record the
voltage transient.
 Repeat the process several times except this time sop the
magnet at various distances inside the solenoid using the ball of
modeling clay. Again record the transient.

Data,
Calculations 
By comparing the recordings in step 4 with
the original trace, it should be possible to identify the position of
the magnet at any points in the voltage where the voltage crosses the
axis an goes from positive to negative or vice versa (flips polarity).
indicate the magnet's position at these points. Write a short
explanation for the transient's appearance noting anything of interest
including any points where the voltage flips polarity. 
Questions,
Conclusions 
Magnets can become demagnetized by
repeated impacts. Take appropriate precautions' to pad the impact of
the magnet when it lands after falling through the solenoid. 
Resources/Materials: 
Solenoid, cow magnet, meter stick, ball of
modeling clay, spacers, computer
system set up with Vernier LabPro software and Lab Pro units 
Essential Question:
Is Lenz's Law a different
form of the first law of thermodynamics? 
Lenz's Law
 There's no free lunch

State the direction of current in a loop of wire
passing through a magnetic field.

State Lenz's Law.

Use Lenz's Law to determine the direction of current
flow in loops of wire with changing magnetic fluxes.
Relevance:
Lenz's Law explains why it's impossible to get more electrical energy out of
a generator than the mechanical energy required to turn it.
Homefun (formative summative
assessment): prob
49

 Drop a magnet down a copper tube and note the time to fall.
 Drop a piece of steel the same size as the magnet down and compare
the time of falling to the first case.
 Note that the magnet drifts downward much more slowly than the
non magnet


Essential Question:
What is the key
difference in the generation of AC vs. DC power? 
Generating Voltages With BFields
 Solve motional EMF problems. For a bar of length L
moving at constant velocity perpendicular to a Bfield:
F_{B} = q v B,
equation 1)
e = work done by F_{B}
per unit of charge
Note that the force on
each charge (electron) within the metal bar moves the charge the distance L
from one end of the bar to the other, hence:
e =
F_{B} (L) / q
Substituting for F_{E }
from equation 1) yields:
e =
q v B (L) / q
= v B L
 Rotating bar (note: v = r
w)
 Sliding bar
(note: v = terminal velocity)

Rotating loop
(note:
F_{B}
= B A cos q)
 Loop
sliding at constant velocity through a constant Bfield (p. 877)
 Use Lenz's Law to calculate forces in motional EMF problems.
Key Principle: mechanical power in = electrical
power out (in other words rate of energy converted to heat by the circuit's
resistance)
Homefun (formative summative
assessment): prob 23,
25, 27

Demo: Hand Cranked
Generator 
 Attach a hand cranked generator to a low voltage light bulb.
 Crank the generator until the bulb glows brightly.
 Crank the generator with nothing attached.
Questions:
 Why is there a sense of resistance when cranking the generator
while attached to the light bulb but not while attached to nothing?
 The generator converts mechanical energy into electrical energy.
Is it 100% efficient and could it ever be?
 How is the energy conversion different with the generator vs.
the process of converting heat to work done by a heat engine?


Formal Physics Investigation 
Title 
Investigation of an AC alternator 
Purpose 
Determine the relationship between the
frequency output and amplitude of an AC alternator spinning at various
rates of rotation. 
Overview 
The modified 5 1/4 inch floppy drive has a
DC motor with and integral AC alternator built into the back of the
motor. The alternator generates a sine wave voltage output, which at
one time was used as a speed control signal for the motor. The
alternator is now attached to a BNC jack that can be connected with a
coaxial cable to an oscilloscope.
 Attach the alternator to the oscilloscope and a variable power
supply to the DC motor.
 Run the motor at various speeds and record the amplitude and
frequency from the oscilloscope.

Data,
Calculations 
Plot the amplitude of the alternator's
output vs. the frequency. 
Questions,
Conclusions 
Why would the frequency be directly
proportional to the alternator's rate of rotation?
Why would the
amplitude of the sine wave increase with the rate of rotation.
Describe the relationship between the amplitude and frequency on
your plot. Is it linear or non linear and why?

Resources/Materials: 
Modified floppy drive, coaxial jumper,
banana plug wires, variable power supplies, oscilloscope 
Essential Question:
Why is it essential to have
mathematical models for wireless communication ? 
Maxwell Equations
 Describe the electric field from an EMF induced by by a magnetic field
and state its general form.
∮E•ds
=  dF_{B}
/ dt
Note: E•ds
yields units of energy divided by charge, in other words, voltage. (See
objective 17. It is just a different form of the above equation.)
 Calculate the electric field generated outside
a solenoid with a radius = R, n coils per unit of length, and a
variable current I = I_{o} cos wt.
Note: if the
Bfield inside a solenoid is changing, it creates an Efield both inside
and outside of the solenoid.

B

=
m_{o }
n I 


=
m_{o }
n I_{o}
cos wt 




∮E•ds 
=

pR^{2}( m_{o }
n I_{o}
w
sin
wt) 

E•2pr 
=
pR^{2}(m_{o }
n I_{o}
w
sin
wt) 

E 
=
(R^{2}m_{o }
n I_{o}
w
sin
wt)
/
(2r) 



 Be as one with the 4 Maxwell equations. Relevance:
The Maxwell equations are the basis for all forms of wireless communication,
one of the key areas in electrical engineering.

Gauss's Law: 
∮E •
da 
= Q
/ε_{o} 

Gauss's Law in Magnetism 
∮B • da 
= 0


Faraday's Law 
∮E •
ds 
=
 dF_{B}
/ dt 

AmpereMaxwell law 
∮B •
ds 
=
m_{o}I
 e_{o}m_{o}dF_{E}
/ dt 
Summative Assessment: Test objectives
121 
Use a dipole antennae connected to a business band radio
transmitter to light a nearby florescent tube without making contact
with it.
Questions:
 Can power be transmitted wirelessly?


AP Physics C E&M Standards
C. Electric circuits
(continued)..................................................................20%
1. Current, resistance, power
2. Steadystate direct current circuits with batteries
and resistors only
3.Capacitors in circuits
a. Steady state
b. Transients in RC circuits *
Essential Question:
What is a capacitor and why should
we care? 
How to Design Giant Capacitors (Chap26 Serway)
Relevance:
Capacitors are key electronic components found in almost any circuit. They
act like springs, storing and releasing electrical energy. Capacitors can reduce
otherwise harmful voltage fluctuations in a circuit.
 Define capacitance mathematically (p. 743).
 C = Q/V
 Calculate capacitance for a parallel plate capacitor (p. 743).
 C = K* e_{o} * A/d
 Calculate the energy stored in a capacitor.
 U = 1/2 *C*V^2

Calculate and describe the Efield in a
capacitor.

Solve capacitor circuit problems.

Solve problems in which dielectric material is inserted or
removed (p.751).
 Battery
attached:
voltage = constant, charge = variable
 Detached from Battery: voltage =
variable, charge = constant
Homefun
(formative summative
assessment): Questions 110 p. 762; prob. 11, 15, 29,
33, 73 p.764769

Video: Demonstration of
Electrostatic Percipitator 
Show video of various capacitor demonstrations. Explain that a
capacitor is an energy storage device like a spring.
Questions:
 Why would a capacitor be useful in power supply designed to
convert AC into DC?
 What type of power do most electronic devices use internally?
 What is the most obvious way to increase the capacitance of a
capacitor without changing the volume of the device. In other words,
without making it large.


Essential Question:
How are resistors, capacitors, and
inductors analogous to elements in mechanics? 
RC Circuits
 State how a capacitor behaves at time = 0 and
infinity.
 Solve RC circuit problems using the above principle
for how capacitors behave at time equals zero and infinity.
 Use the above principles to sketch the following
curve for a charging capacitor:
 charge vs time
 current vs time
 Use the above principles to sketch the following
curve for a discharging capacitor:
 charge vs time
 current vs time
 voltage vs time
 Using Kirchhoff's Law, write the differential equation for an RC
circuit (p.808).
e = q / C
for a capacitor
e 
iR  q / C = 0
for an RC circuit
Ce
 (dq/dt) RC  q = 0
RC(dq/dt) + q  Ce
= 0
 For a charging RC circuit (p.808) Calculate the
following:
 charge vs time
 current vs time
 time constant
 For a discharging RC circuit (p.808) Calculate the
following:
 charge vs time
 current vs time
 voltage vs time
 time constant
 Describe how time constant could be used to measure
the capacitance of an unknown capacitor.
t = RC
Homefun (formative summative
assessment): prob. 43,
44, 45 p. 824
Formal Physics Investigation 
Title 
Investigation of the Discharging Curve for
a Capacitor 
Purpose 
To estimate the capacitance of a capacitor
using time constant and a voltage drop vs time curve for capacitor
discharging through a known resistor. 
Models 
t
= RC, v = v_{o}
e ^{t / RC} 
Overview 
 Turn the power supply to its lowest voltage setting. (We will be
using only DC voltage).
 With the power supply off, connect the positive terminal of the
capacitor to a resistor and the resistor to the positive terminal of
the power supply. Complete the circuit by connecting the negative
terminal of the power supply to the negative terminal of the
capacitor.
 Connect the Vernier voltage probes to the capacitor just like a
voltmeter.
 Turn on the power supply and charge the capacitor to 6 volts
while collecting data on the charging curve.
 Disconnect the power supply and discharge the capacitor through
the resistor while collecting data with the Vernier system.

Safety
Issues 
Unlike batteries, capacitors can discharge
their energy almost instantaneously. Shorting one out will very likely
result in equipment damage and possibly a fire.
Capacitors must be
both charged and discharged through a resistor or damage will result. 
Equipment
Limitations 
Do not exceed 10 volts. Higher voltages
may damage the LabPro's electronics 
Resources/Materials: 
Large sized capacitor, resistor of known
value, computer
system set up with Vernier LabPro software and Lab Pro units, wires,
power supply 
AP Physics C E&M Standards
E. Electromagnetism
(continued)............................................................16%
1. Electromagnetic induction (including Faraday's
law and Lenz's law)
2. Inductance (including LR and LC circuits) *
3. Maxwell's equations *
LR Circuits (Chap. 32 Serway)
Relevance: Inductors are a key element in numerous types of electronic devices
including computer power supplies.
AP Physics C E&M Standards 
E. Electromagnetism (16 %), 2. Inductance
(including LR circuits)
 State how an inductor behaves at time = 0 and
infinity.
 Solve LR circuit problems using the above principle
for how inductors behave at time equals zero and infinity.
 Use the above principles to sketch the current vs
time for a charging inductor.
 Use the above principles to sketch the current vs
time and voltage vs time curve for a discharging inductor:
 For a charging LR circuit (p.944) Calculate the following:
 Note:
 e_{L}
=  L (di/dt)
Using
Kirchhoff's Law
e

L (di/dt)  Ri = 0
L (di/dt) + Ri
 e
= 0


 For a discharging LR circuit (p.944) Calculate the following:
 current vs time
 voltage vs time
 time constant
 Describe how time constant could be used to measure
the inductance of an unknown inductor.
t = L / R
 Calculate the energy stored in an inductor.
U_{L} = 1/2 L I^{2}
Homefun (formative summative
assessment): 17, 19, 21 p. 957

Hollywood Video Clip: The Core 
Show a video clip of the Virgil's crew communicating with the
surface from deep within the Earth.
Questions:
 How could a vehicle communicate with the surface from inside the
Earth?
 Is there such a thing as a radio controlled submarine?
The Skin Depth Equation (Wikipedia)
The
current density J in an infinitely thick plane
conductor
decreases exponentially with depth δ from the surface, as
follows:

where d is a constant called the skin depth. This is
defined as the depth below the surface of the conductor at which the
current density decays to 1/e
(about 0.37) of the current density at the surface (J_{S}).
It can be calculated as follows:

where
 ρ =
resistivity of conductor
 ω =
angular frequency of current = 2π × frequency
 μ = absolute
magnetic permeability of conductor
,
where μ_{0} is the permeability
of free space and μ_{r}
is the relative permeabilty of the conductor.


Essential Question:
Can an electrical circuit resonate
and why would this be important? 
LC and RLC Circuits (Chap. 32
Serway)
 Write an energy balance equation for an LC circuit.
(see the Physics
of Resonance  Electrical Circuits, also see p. 949 Serway)) Relevance:
Resonating LC circuits are the basis of wireless communication and
electronic music.
U = U _{C} + U_{L}
 Calculate the frequency of an LC circuit.
U = U _{C} + U_{L}
U = Q^{2}/(2C) + i^{2 }
(L / 2)
but
Q_{max}^{2}/(2C)
= i_{max}^{2 }
(L / 2)
Q_{max}^{2}/(2C)
= ( ωQ_{max})^{2 }
(L / 2)
ω = [ 1 / (LC) ]^{0.5}
f = 1 / [2p
(LC) ^{0.5 }] 

Q = Q_{max}(cos ωt)
dQ/dt =
 ωQ_{max}(sin ωt)
i =
 ωQ_{max}(sin ωt)
i_{max}
=  ωQ_{max} 
 Draw an analogy between an LC circuit and a spring
and mass system.
 Draw an analogy between an RLC circuit and a spring
and mass system.
Electrical 
Mechanical 
Capacitance C 
(1/C) q 
Spring 
k x 
Resistance R 
R dq/dt 
Viscous damper 
b dx/dt 
Inductance L 
L (d^{2}q/dt^{2}) 
Inertia 
m (d^{2}x/dt^{2}) 
 Explain the difference between dampening and
damping.
Summative Assessment: Test
objectives 126 
Demonstration from
The Physics of
Resonance.
 Briefly explain how a crystal radio works.
 Connect a crystal radio to an oscilloscope. Inductively couple a
DC power supply to the crystal radio and give it a pulse.
 Observe the decaying sine wave.
The crystal radio is configured to resonate at the radio station's
frequency it is tuned to. Subjecting the radio to a pulse is like
striking a bell: these respective action cause the both the bell and
the electrical circuits to resonate with the bell emitting sound at
its natural frequencies and the radio emitting radio waves.
Questions:
 Why does an extremely strong electromagnetic pulse of energy,
such as would be produced by an atomic bomb tend to wipe out
wireless communications?


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