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AP · AP C:E&M · free response

AP Physics C: Electricity and Magnetism · Question 10

AP Physics C: Electricity and Magnetism · Original GioPhysics question with a detailed, mark-by-mark answer guide.

Demand
discriminating
Marks
14
Topics
1
Answer
Complete
A conducting rod sliding on two rails closed by a resistor, seen from aboveRconducting rod, mass mrailsv₀Luniform magnetic field B, out of the page
Fig. 10.1The apparatus is seen from above. Two long parallel horizontal rails run across the figure, joined at their left-hand ends by a resistor labelled R, and open at the right-hand end. A short bar lying across the rails, drawn solid, is the conducting rod of mass m. An arrow starting at the rod points to the right and is labelled v0. A tick-marked dimension line drawn between the rails to the right of the rod is labelled L for their separation. Dots in an even grid cover the whole area, marking a uniform magnetic field pointing out of the page, that is vertically upward from the horizontal plane of the rails. No induced current or force is drawn.
free response14 marks

A conducting rod of mass m and length L slides without friction on two long horizontal parallel rails separated by L. The rails are joined at one end by a resistor of resistance R, and the whole apparatus sits in a uniform magnetic field of magnitude B directed vertically, perpendicular to the plane of the rails. The rod is given an initial speed v₀ to the right at t = 0 and then released. The rod and rails have negligible resistance.

  1. (a)

    Derive Derive expressions for the e.m.f. induced in the circuit and the current in the resistor when the rod moves with speed v.

    3 marks
  2. (b)

    Derive Derive an expression for the speed of the rod as a function of time.

    4 marks
  3. (c)

    Determine Determine the total energy dissipated in the resistor from t = 0 until the rod has effectively stopped, and determine the total distance the rod travels.

    3 marks
  4. (d)

    Explain Explain, without deriving anything further, why the rod slows down, and explain what would be observed instead if the resistor were replaced by a superconducting wire of zero resistance.

    4 marks
Ready to self-mark?Reveal the detailed answer guide
Answer overviewKey answer: 1 point: the flux through the circuit changes as the enclosed area changes, ε = −dΦ/dt = −BL(dx/dt) 1 point: magnitude of the e.m.f. is BLv 1 point: I = BLv/R
01

(a)

3 marks

Derive Derive expressions for the e.m.f. induced in the circuit and the current in the resistor when the rod moves with speed v.

How to approach it

List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.

  1. 1

    1 point: the flux through the circuit changes as the enclosed area changes, ε = −dΦ/dt = −BL(dx/dt)

  2. 2

    1 point: magnitude of the e.m.f. is BLv

  3. 3

    1 point: I = BLv/R

02

(b)

4 marks

Derive Derive an expression for the speed of the rod as a function of time.

How to approach it

List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.

  1. 1

    1 point: the force on the current-carrying rod is F = BIL = B²L²v/R, directed opposite to the motion by Lenz's law

  2. 2

    1 point: m dv/dt = −B²L²v/R

  3. 3

    1 point: separates and integrates from v₀ at t = 0

  4. 4

    1 point: v(t) = v₀e^(−B²L²t/(mR))

03

(c)

3 marks

Determine Determine the total energy dissipated in the resistor from t = 0 until the rod has effectively stopped, and determine the total distance the rod travels.

How to approach it

List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.

  1. 1

    1 point: the only energy source is the rod's kinetic energy, and it all ends up in the resistor, so the total dissipated energy is ½mv₀²

  2. 2

    1 point: distance x = ∫₀^∞ v dt = v₀ mR/(B²L²)

  3. 3

    1 point: states both results with correct symbols

04

(d)

4 marks

Explain Explain, without deriving anything further, why the rod slows down, and explain what would be observed instead if the resistor were replaced by a superconducting wire of zero resistance.

How to approach it

State the outcome first, then link cause to effect with the relevant physical principle. Each link in the reasoning should be explicit enough to earn its own marking point.

  1. 1

    1 point: the motion changes the flux through the circuit, which drives a current

  2. 2

    1 point: by Lenz's law that current is in the direction whose magnetic force opposes the motion, so the rod decelerates — the kinetic energy is transferred to the resistor as thermal energy

  3. 3

    1 point: with zero resistance the current, and hence the retarding force, would become arbitrarily large for any non-zero speed

  4. 4

    1 point: so the rod would stop almost immediately, and in the idealised limit the flux through the circuit cannot change at all — which is the behaviour that makes a superconducting loop hold its flux fixed

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