Mr. Rogers' AP Physics C: E&M (with IB Physics) Objectives

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3rd Q objectives small investigations IB internal assessment write up specs IB rubrics  
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?


  1. 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 = mo I

  1. 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 =   mo N I



  solenoid: B =   mo N I



  current carrying wire: B =   mo I r



Homefun Prob 23, 25 p.895


Essential Question: How can a magnetic field be used to generate electricity?

Magnetic Flux

  1. Mathematically define magnetic flux--roughly speaking, the amount of magnetic field passing through a given surface.

FB = BdA  

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.

  1. Explain briefly how a transformer works and why it requires an AC input.
  2. Nout / Nin = Vout / Vin

  3. Calculate the magnetic flux through a rectangular loop of wire next to a long thin current carrying wire in the same plane (p.850).

    Magnetic flux through a loop above a current carrying wire

    FB= [ ( mo I b) / 2p ] ln (1 + a/c)


  1. State Gauss's law for magnetism. Why is the magnetic flux through a closed surface zero? Because, unlike E-fields, 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.

BdA = 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.

Demo: Tesla Coil

Use a Tesla coil to light up a florescent tube from a distance. 


  1. 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)

  1. 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 = - dFB / 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.
  1. Connect the terminals of the solenoid to the the LabPro voltage probe.
  2. Place the tip of the magnet at the entrance to the solenoid
  3. Drop the magnet completely through the solenoid and record the voltage transient.
  4. 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

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

  2. State Lenz's Law.

  3. 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



Demo: Lenz's Law

  1. Drop a magnet down a copper tube and note the time to fall.
  2. Drop a piece of steel the same size as the magnet down and compare the time of falling to the first case.
  3. 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 B-Fields

  1. Solve motional EMF problems. For a bar of length L moving at constant velocity perpendicular to a B-field:

FB  = q v B,      equation 1)

e = work done by FB 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 = FB (L) / q

Substituting for FE 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: FB = B A cos q)
      • Loop sliding at constant velocity through a constant B-field (p. 877)
  1. 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

  1. Attach a hand cranked generator to a low voltage light bulb.
  2. Crank the generator until the bulb glows brightly.
  3. Crank the generator with nothing attached.


  1. Why is there a sense of resistance when cranking the generator while attached to the light bulb but not while attached to nothing?
  2. The generator converts mechanical energy into electrical energy. Is it 100% efficient and could it ever be?
  3. 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.
  1. Attach the alternator to the oscilloscope and a variable power supply to the DC motor.
  2. 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

  1. Describe the electric field from an EMF induced by by a magnetic field and state its general form.

Eds =   - dFB / dt

Note: Eds yields units of energy divided by charge, in other words, voltage. (See objective 17. It is just a different form of the above equation.)

  1. Calculate the electric field generated outside a solenoid with a radius = R, n coils per unit of length, and a variable current I = Io cos wt.

Note: if the B-field inside a solenoid is changing, it creates an E-field both inside and outside of the solenoid.


 = mo n I
     = mo n Io cos wt


 = - pR2(- mo n Io w sin wt)


 = pR2(mo n Io w sin wt)


 = (R2mo n Io w sin wt) / (2r)
  1. 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 /εo
  Gauss's Law in Magnetism B da = 0
  Faraday's Law E ds = - dFB / dt
  Ampere-Maxwell law B ds = moI - eomodFE / dt

Summative Assessment: Test objectives 1-21

Demo: Dipole Antennae

Use a dipole antennae connected to a business band radio transmitter to light a nearby florescent tube without making contact with it.


  1. Can power be transmitted wirelessly?


AP Physics C E&M Standards

 C. Electric circuits (continued)..................................................................20%

1. Current, resistance, power
2. Steady-state 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.

  1. Define capacitance mathematically (p. 743).
C = Q/V
  1. Calculate capacitance for a parallel plate capacitor (p. 743).
C = K* eo * A/d
  1. Calculate the energy stored in a capacitor.
U = 1/2 *C*V^2
  1. Calculate and describe the E-field in a capacitor.

  2. Solve capacitor circuit problems.

  3. 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 1-10 p. 762; prob. 11, 15, 29, 33, 73 p.764-769


Video: Demonstration of Electrostatic Percipitator

Show video of various capacitor demonstrations. Explain that a capacitor is an energy storage device like a spring.


  1. Why would a capacitor be useful in power supply designed to convert AC into DC?
  2. What type of power do most electronic devices use internally?
  3. 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

  1. State how a capacitor behaves at time = 0 and infinity.
  2. Solve RC circuit problems using the above principle for how capacitors behave at time equals zero and infinity.
  3. Use the above principles to sketch the following curve for a charging capacitor:
  1. Use the above principles to sketch the following curve for a discharging capacitor:
  1. 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  

  1. For a charging RC circuit (p.808) Calculate the following:
  1. For a discharging RC circuit (p.808) Calculate the following:
  1. 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 = vo e -t / RC
  1. Turn the power supply to its lowest voltage setting. (We will be using only DC voltage).
  2. 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.
  3. Connect the Vernier voltage probes to the capacitor just like a voltmeter.
  4. Turn on the power supply and charge the capacitor to 6 volts while collecting data on the charging curve.
  5. 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)

  1. State how an inductor behaves at time = 0 and infinity.
  2. Solve LR circuit problems using the above principle for how inductors behave at time equals zero and infinity.
  3. Use the above principles to sketch the current vs time for a charging inductor.
  4. Use the above principles to sketch the current vs time and voltage vs time curve for a discharging inductor:
  5. For a charging LR circuit (p.944) Calculate the following:
    • current vs time
    • time constant
eL = - L (di/dt)

Using Kirchhoff's Law

e  - L (di/dt) - Ri = 0

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

  1. For a discharging LR circuit (p.944) Calculate the following:
    • current vs time
    • voltage vs time
    • time constant
  1. Describe how time constant could be used to measure the inductance of an unknown inductor.

t = L / R

  1. Calculate the energy stored in an inductor.

UL = 1/2 L I2

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.


  1. How could a vehicle communicate with the surface from inside the Earth?
  2. 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:

J=J_\mathrm{S} \,e^{-{\delta /d}}

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 (JS). It can be calculated as follows:

d=\sqrt{{2\rho }\over{\omega\mu}}


ρ = resistivity of conductor
ω = angular frequency of current = 2π frequency
μ = absolute magnetic permeability of conductor = \mu_0 \cdot \mu_r, where μ0 is the permeability of free space and μr 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)

  1. 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 + UL

  1. Calculate the frequency of an LC circuit.
U = U C + UL

U = Q2/(2C) + i2 (L / 2)


Qmax2/(2C) = imax2 (L / 2)

Qmax2/(2C) = (- ωQmax)2 (L / 2)

ω = [ 1 / (LC) ]0.5

f = 1 / [2p (LC) 0.5 ]
Q = Qmax(cos ωt)

dQ/dt = - ωQmax(sin ωt)

i = - ωQmax(sin ωt)

imax = - ωQmax

  1. Draw an analogy between an LC circuit and a spring and mass system.
  2. Draw an analogy between an RLC circuit and a spring and mass system.


Capacitance   C  (1/C) q Spring  k x
Resistance     R  R dq/dt Viscous damper b dx/dt
Inductance      L  L (d2q/dt2) Inertia m (d2x/dt2)
  1. Explain the difference between dampening and damping.


Summative Assessment: Test objectives 1-26

Demo: Crystal Radio

Demonstration from  The Physics of Resonance.

  1. Briefly explain how a crystal radio works.
  2. Connect a crystal radio to an oscilloscope. Inductively couple a DC power supply to the crystal radio and give it a pulse.
  3. 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.


  1. 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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