Electromagnetism Free Response Practice Questions | AP Physics C Electricity and Magnetism - Grade 9 PDF Download

 Page 1


Section II: Free Response
1. The diagram below shows two views of a metal rod of length l rotating with constant angular
speed ? about an axis that is in the plane of the page. The rotation takes place in a uniform
magnetic field B whose direction is parallel to the angular velocity ?.
(a) What is the emf induced between the ends of the rod?
(b) What is the polarity (+ or -) of the rotating end?
In the following diagram, a metal rod of length l moves with constant velocity v parallel to a
long, straight wire carrying a steady current I. The lower end of the rod maintains a distance
of a from the straight wire.
(c) What is the emf induced between the ends of the rod?
(d) What is the polarity (+ or -) of the end that is farther from the straight wire?
2. A rectangular loop of wire (side lengths a and b) rotates with constant angular speed ? in a
uniform magnetic field B. At time t = 0, the plane of the loop is perpendicular to B, as shown
in the figure on the left. The magnetic field B is directed to the right (in the +xdirection), and
Page 2


Section II: Free Response
1. The diagram below shows two views of a metal rod of length l rotating with constant angular
speed ? about an axis that is in the plane of the page. The rotation takes place in a uniform
magnetic field B whose direction is parallel to the angular velocity ?.
(a) What is the emf induced between the ends of the rod?
(b) What is the polarity (+ or -) of the rotating end?
In the following diagram, a metal rod of length l moves with constant velocity v parallel to a
long, straight wire carrying a steady current I. The lower end of the rod maintains a distance
of a from the straight wire.
(c) What is the emf induced between the ends of the rod?
(d) What is the polarity (+ or -) of the end that is farther from the straight wire?
2. A rectangular loop of wire (side lengths a and b) rotates with constant angular speed ? in a
uniform magnetic field B. At time t = 0, the plane of the loop is perpendicular to B, as shown
in the figure on the left. The magnetic field B is directed to the right (in the +xdirection), and
the rotation axis is the y-axis (with ? in the +y direction), and the four corners of the loop are
labeled 1, 2, 3, and 4. (Express answers in terms of a, b, ?, B, and fundamental constants.)
(a) Find a formula that gives the magnetic flux B through the loop as a function of time, t.
(b) Find a formula that gives the emf induced in the loop as a function of time, t.
(c) If the total resistance of the loop is R, what is the current induced in the loop?
(d) When ?t = p/2, is the induced current in the loop directed from Point 1 to Point 2
(-ydirection) or from Point 2 to Point 1 (+y direction)?
(e) Find the rate at which energy is dissipated (as joule heat) in the wires that comprise the
loop, and the amount of energy dissipated per revolution.
(f) Find the external torque required to keep the loop rotating at the constant angular speed ?.
3. The figure below shows a toroidal solenoid of mean radius R and N total windings. The cross-
sections of the toroid are circles of radius a (which is much smaller than R, so variations in the
magnetic field strength within the space enclosed by the windings may be neglected).
Page 3


Section II: Free Response
1. The diagram below shows two views of a metal rod of length l rotating with constant angular
speed ? about an axis that is in the plane of the page. The rotation takes place in a uniform
magnetic field B whose direction is parallel to the angular velocity ?.
(a) What is the emf induced between the ends of the rod?
(b) What is the polarity (+ or -) of the rotating end?
In the following diagram, a metal rod of length l moves with constant velocity v parallel to a
long, straight wire carrying a steady current I. The lower end of the rod maintains a distance
of a from the straight wire.
(c) What is the emf induced between the ends of the rod?
(d) What is the polarity (+ or -) of the end that is farther from the straight wire?
2. A rectangular loop of wire (side lengths a and b) rotates with constant angular speed ? in a
uniform magnetic field B. At time t = 0, the plane of the loop is perpendicular to B, as shown
in the figure on the left. The magnetic field B is directed to the right (in the +xdirection), and
the rotation axis is the y-axis (with ? in the +y direction), and the four corners of the loop are
labeled 1, 2, 3, and 4. (Express answers in terms of a, b, ?, B, and fundamental constants.)
(a) Find a formula that gives the magnetic flux B through the loop as a function of time, t.
(b) Find a formula that gives the emf induced in the loop as a function of time, t.
(c) If the total resistance of the loop is R, what is the current induced in the loop?
(d) When ?t = p/2, is the induced current in the loop directed from Point 1 to Point 2
(-ydirection) or from Point 2 to Point 1 (+y direction)?
(e) Find the rate at which energy is dissipated (as joule heat) in the wires that comprise the
loop, and the amount of energy dissipated per revolution.
(f) Find the external torque required to keep the loop rotating at the constant angular speed ?.
3. The figure below shows a toroidal solenoid of mean radius R and N total windings. The cross-
sections of the toroid are circles of radius a (which is much smaller than R, so variations in the
magnetic field strength within the space enclosed by the windings may be neglected).
(a) Use Ampere’s law to find the magnetic field strength within the toroid. Write your answer in
terms of N, I, R, and fundamental constants.
A circular loop of wire of radius 2a is placed around the toroid as shown:
Assume that the current in the toroid is varied sinusoidally according to the equation I(t)
= I
0
 sin ?t, where I
0
 and ? are fixed constants.
(b) Determine the emf induced in the circular wire loop.
(c) Determine the electric field induced at the position of the circular wire loop.
(d) What is the self-inductance of the toroidal solenoid?
4. A circuit is connected as shown above. The switch S is initially open. Then it is moved to
position A.
Page 4


Section II: Free Response
1. The diagram below shows two views of a metal rod of length l rotating with constant angular
speed ? about an axis that is in the plane of the page. The rotation takes place in a uniform
magnetic field B whose direction is parallel to the angular velocity ?.
(a) What is the emf induced between the ends of the rod?
(b) What is the polarity (+ or -) of the rotating end?
In the following diagram, a metal rod of length l moves with constant velocity v parallel to a
long, straight wire carrying a steady current I. The lower end of the rod maintains a distance
of a from the straight wire.
(c) What is the emf induced between the ends of the rod?
(d) What is the polarity (+ or -) of the end that is farther from the straight wire?
2. A rectangular loop of wire (side lengths a and b) rotates with constant angular speed ? in a
uniform magnetic field B. At time t = 0, the plane of the loop is perpendicular to B, as shown
in the figure on the left. The magnetic field B is directed to the right (in the +xdirection), and
the rotation axis is the y-axis (with ? in the +y direction), and the four corners of the loop are
labeled 1, 2, 3, and 4. (Express answers in terms of a, b, ?, B, and fundamental constants.)
(a) Find a formula that gives the magnetic flux B through the loop as a function of time, t.
(b) Find a formula that gives the emf induced in the loop as a function of time, t.
(c) If the total resistance of the loop is R, what is the current induced in the loop?
(d) When ?t = p/2, is the induced current in the loop directed from Point 1 to Point 2
(-ydirection) or from Point 2 to Point 1 (+y direction)?
(e) Find the rate at which energy is dissipated (as joule heat) in the wires that comprise the
loop, and the amount of energy dissipated per revolution.
(f) Find the external torque required to keep the loop rotating at the constant angular speed ?.
3. The figure below shows a toroidal solenoid of mean radius R and N total windings. The cross-
sections of the toroid are circles of radius a (which is much smaller than R, so variations in the
magnetic field strength within the space enclosed by the windings may be neglected).
(a) Use Ampere’s law to find the magnetic field strength within the toroid. Write your answer in
terms of N, I, R, and fundamental constants.
A circular loop of wire of radius 2a is placed around the toroid as shown:
Assume that the current in the toroid is varied sinusoidally according to the equation I(t)
= I
0
 sin ?t, where I
0
 and ? are fixed constants.
(b) Determine the emf induced in the circular wire loop.
(c) Determine the electric field induced at the position of the circular wire loop.
(d) What is the self-inductance of the toroidal solenoid?
4. A circuit is connected as shown above. The switch S is initially open. Then it is moved to
position A.
(a) Determine the current in the circuit immediately after the switch is closed.
(b) Determine the current in the circuit a long time after the switch is closed.
Some time after the steady state situation has been reached, the switch is moved almost
instantaneously from position A to position B.
(c) Determine the current through the 5 O resistor immediately after the switch has been
moved.
(d) Determine the potential difference across the inductor immediately after the switch has
been closed.
Page 5


Section II: Free Response
1. The diagram below shows two views of a metal rod of length l rotating with constant angular
speed ? about an axis that is in the plane of the page. The rotation takes place in a uniform
magnetic field B whose direction is parallel to the angular velocity ?.
(a) What is the emf induced between the ends of the rod?
(b) What is the polarity (+ or -) of the rotating end?
In the following diagram, a metal rod of length l moves with constant velocity v parallel to a
long, straight wire carrying a steady current I. The lower end of the rod maintains a distance
of a from the straight wire.
(c) What is the emf induced between the ends of the rod?
(d) What is the polarity (+ or -) of the end that is farther from the straight wire?
2. A rectangular loop of wire (side lengths a and b) rotates with constant angular speed ? in a
uniform magnetic field B. At time t = 0, the plane of the loop is perpendicular to B, as shown
in the figure on the left. The magnetic field B is directed to the right (in the +xdirection), and
the rotation axis is the y-axis (with ? in the +y direction), and the four corners of the loop are
labeled 1, 2, 3, and 4. (Express answers in terms of a, b, ?, B, and fundamental constants.)
(a) Find a formula that gives the magnetic flux B through the loop as a function of time, t.
(b) Find a formula that gives the emf induced in the loop as a function of time, t.
(c) If the total resistance of the loop is R, what is the current induced in the loop?
(d) When ?t = p/2, is the induced current in the loop directed from Point 1 to Point 2
(-ydirection) or from Point 2 to Point 1 (+y direction)?
(e) Find the rate at which energy is dissipated (as joule heat) in the wires that comprise the
loop, and the amount of energy dissipated per revolution.
(f) Find the external torque required to keep the loop rotating at the constant angular speed ?.
3. The figure below shows a toroidal solenoid of mean radius R and N total windings. The cross-
sections of the toroid are circles of radius a (which is much smaller than R, so variations in the
magnetic field strength within the space enclosed by the windings may be neglected).
(a) Use Ampere’s law to find the magnetic field strength within the toroid. Write your answer in
terms of N, I, R, and fundamental constants.
A circular loop of wire of radius 2a is placed around the toroid as shown:
Assume that the current in the toroid is varied sinusoidally according to the equation I(t)
= I
0
 sin ?t, where I
0
 and ? are fixed constants.
(b) Determine the emf induced in the circular wire loop.
(c) Determine the electric field induced at the position of the circular wire loop.
(d) What is the self-inductance of the toroidal solenoid?
4. A circuit is connected as shown above. The switch S is initially open. Then it is moved to
position A.
(a) Determine the current in the circuit immediately after the switch is closed.
(b) Determine the current in the circuit a long time after the switch is closed.
Some time after the steady state situation has been reached, the switch is moved almost
instantaneously from position A to position B.
(c) Determine the current through the 5 O resistor immediately after the switch has been
moved.
(d) Determine the potential difference across the inductor immediately after the switch has
been closed.
Section II: Free Response
1. (a) Consider a small section of length dx of the rod at a distance x from the nonrotating end.
The velocity of this small piece is v = ?x, so the motional emf induced between its ends
is de = vB dx = ?xB dx. Integrating this from x = 0 to x = l gives
(b) Refer to the view from above of the rod:
By the right-hand rule, the direction of v × B is to the right, so the magnetic force on free
electrons will be to the left, leaving excess positive charge at the right (rotating) end.
(c) Consider a small section of length dy at a distance y from the straight wire.
Since the magnetic field at the position of this section is (µ
0
/2p)(I/y), the motional emf
induced between its ends is de = vB dy = v(µ
0
/2p)(I/y) dy. Integrating this
from y = ato y = a +  gives