Magnetic Fields 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 a simple mass spectrograph. It consists of a source of ions (charged
atoms) that are accelerated (essentially from rest) by the voltage V and enter a region
containing a uniform magnetic field, B. The polarity of V may be reversed so that both
positively charged ions (cations) and negatively charged ions (anions) can be accelerated.
Once the ions enter the magnetic field, they follow a semicircular path and strike the front wall
of the spectrograph, on which photographic plates are constructed to record the impact.
(a) What is the acceleration of an ion of charge q just before it enters the magnetic field?
(b) Find the speed with which an ion of charge q enters the magnetic field.
(c) (i) Which semicircular path, 1 or 2, would a cation follow?
(ii) Which semicircular path, 1 or 2, would an anion follow?
(d) Determine the mass of a cation entering the apparatus in terms of y, q, B, and V.
(d) Once a cation of charge q enters the magnetic field, how long does it take to strike the
photographic plate?
(f) What is the work done by the magnetic force in the spectrograph on a cation of charge q?
2. A wire of diameter d and resistivity ? is bent into a rectangular loop (of side lengths a and b)
and fitted with a small battery that provides a voltage V. The loop is placed at a distance cfrom
a very long, straight wire that carries a current I in the direction indicated in the diagram.
Page 2


Section II: Free Response
1. The diagram below shows a simple mass spectrograph. It consists of a source of ions (charged
atoms) that are accelerated (essentially from rest) by the voltage V and enter a region
containing a uniform magnetic field, B. The polarity of V may be reversed so that both
positively charged ions (cations) and negatively charged ions (anions) can be accelerated.
Once the ions enter the magnetic field, they follow a semicircular path and strike the front wall
of the spectrograph, on which photographic plates are constructed to record the impact.
(a) What is the acceleration of an ion of charge q just before it enters the magnetic field?
(b) Find the speed with which an ion of charge q enters the magnetic field.
(c) (i) Which semicircular path, 1 or 2, would a cation follow?
(ii) Which semicircular path, 1 or 2, would an anion follow?
(d) Determine the mass of a cation entering the apparatus in terms of y, q, B, and V.
(d) Once a cation of charge q enters the magnetic field, how long does it take to strike the
photographic plate?
(f) What is the work done by the magnetic force in the spectrograph on a cation of charge q?
2. A wire of diameter d and resistivity ? is bent into a rectangular loop (of side lengths a and b)
and fitted with a small battery that provides a voltage V. The loop is placed at a distance cfrom
a very long, straight wire that carries a current I in the direction indicated in the diagram.
(Express all answers in terms of a, b, c, d, ?, V, I, m, B, x, and fundamental constants.)
(a) When the switch S is closed, find the current in the rectangular loop.
(b) What is the magnetic force (magnitude and direction) exerted on the loop by the long,
straight wire?
(c) The wire of the rectangular loop is then reshaped into a circle. What will be the radius of the
circular loop?
(d) If the loop constructed in part (c) were then threaded around the long, straight wire (so that
the straight wire passed through the center of the circular loop), what would be the
magnetic force on the loop now?
(e) In the following diagram, two fixed L-shaped wires, separated by a distance x, are connected
by a wire that’s free to slide vertically.
The mass of the sliding wire, S, is m. If the sliding wire S crosses a region that contains a
uniform magnetic field B, how much current must be carried by the wire to keep S from
sliding down (due to its weight)?
3. The figure below shows two long, straight wires connected by a circular arc of radius x that
subtends a central angle ?. The current in the wire is I.
Page 3


Section II: Free Response
1. The diagram below shows a simple mass spectrograph. It consists of a source of ions (charged
atoms) that are accelerated (essentially from rest) by the voltage V and enter a region
containing a uniform magnetic field, B. The polarity of V may be reversed so that both
positively charged ions (cations) and negatively charged ions (anions) can be accelerated.
Once the ions enter the magnetic field, they follow a semicircular path and strike the front wall
of the spectrograph, on which photographic plates are constructed to record the impact.
(a) What is the acceleration of an ion of charge q just before it enters the magnetic field?
(b) Find the speed with which an ion of charge q enters the magnetic field.
(c) (i) Which semicircular path, 1 or 2, would a cation follow?
(ii) Which semicircular path, 1 or 2, would an anion follow?
(d) Determine the mass of a cation entering the apparatus in terms of y, q, B, and V.
(d) Once a cation of charge q enters the magnetic field, how long does it take to strike the
photographic plate?
(f) What is the work done by the magnetic force in the spectrograph on a cation of charge q?
2. A wire of diameter d and resistivity ? is bent into a rectangular loop (of side lengths a and b)
and fitted with a small battery that provides a voltage V. The loop is placed at a distance cfrom
a very long, straight wire that carries a current I in the direction indicated in the diagram.
(Express all answers in terms of a, b, c, d, ?, V, I, m, B, x, and fundamental constants.)
(a) When the switch S is closed, find the current in the rectangular loop.
(b) What is the magnetic force (magnitude and direction) exerted on the loop by the long,
straight wire?
(c) The wire of the rectangular loop is then reshaped into a circle. What will be the radius of the
circular loop?
(d) If the loop constructed in part (c) were then threaded around the long, straight wire (so that
the straight wire passed through the center of the circular loop), what would be the
magnetic force on the loop now?
(e) In the following diagram, two fixed L-shaped wires, separated by a distance x, are connected
by a wire that’s free to slide vertically.
The mass of the sliding wire, S, is m. If the sliding wire S crosses a region that contains a
uniform magnetic field B, how much current must be carried by the wire to keep S from
sliding down (due to its weight)?
3. The figure below shows two long, straight wires connected by a circular arc of radius x that
subtends a central angle ?. The current in the wire is I.
(a) Find the magnetic field (magnitude and direction) created at Point C. Write your answer in
terms of x, ?, I, and fundamental constants.
(b) A particle of charge +q is placed at Point C and released. Find the magnetic force on the
particle.
(c) A second long, straight wire is set up perpendicular to the plane of the page through C,
carrying the same current, I (directed out of the page), as the wire pictured in the diagram.
Determine the magnetic force per unit length between the wires.
4. For a conducting rod that carries a current I, the current density is defined as the current per
unit area: J = I/A.
Part 1. A homogeneous cylindrical rod of radius R carries a current whose current density, J,
is uniform (constant); that is, J does not vary with the radial distance, r, from the center of the
rod.
(a) Determine the total current, I, in the rod.
(b) Calculate the magnitude of the magnetic field for
(i) r < R
(ii) r > R, writing your answers in terms of r, R, I, and fundamental constants
Part 2. A nonhomogeneous cylindrical rod of radius R carries a current whose current
density, J, varies with the radial distance, r, from the center of the rod according to the
equation J = s r, where s is a constant.
(c) What are the units of s ?
(d) Determine the total current, I, in the rod.
(e) Calculate the magnitude of the magnetic field for
(i) r < R
(ii) r > R, writing your answers in terms of r, R, and I
Page 4


Section II: Free Response
1. The diagram below shows a simple mass spectrograph. It consists of a source of ions (charged
atoms) that are accelerated (essentially from rest) by the voltage V and enter a region
containing a uniform magnetic field, B. The polarity of V may be reversed so that both
positively charged ions (cations) and negatively charged ions (anions) can be accelerated.
Once the ions enter the magnetic field, they follow a semicircular path and strike the front wall
of the spectrograph, on which photographic plates are constructed to record the impact.
(a) What is the acceleration of an ion of charge q just before it enters the magnetic field?
(b) Find the speed with which an ion of charge q enters the magnetic field.
(c) (i) Which semicircular path, 1 or 2, would a cation follow?
(ii) Which semicircular path, 1 or 2, would an anion follow?
(d) Determine the mass of a cation entering the apparatus in terms of y, q, B, and V.
(d) Once a cation of charge q enters the magnetic field, how long does it take to strike the
photographic plate?
(f) What is the work done by the magnetic force in the spectrograph on a cation of charge q?
2. A wire of diameter d and resistivity ? is bent into a rectangular loop (of side lengths a and b)
and fitted with a small battery that provides a voltage V. The loop is placed at a distance cfrom
a very long, straight wire that carries a current I in the direction indicated in the diagram.
(Express all answers in terms of a, b, c, d, ?, V, I, m, B, x, and fundamental constants.)
(a) When the switch S is closed, find the current in the rectangular loop.
(b) What is the magnetic force (magnitude and direction) exerted on the loop by the long,
straight wire?
(c) The wire of the rectangular loop is then reshaped into a circle. What will be the radius of the
circular loop?
(d) If the loop constructed in part (c) were then threaded around the long, straight wire (so that
the straight wire passed through the center of the circular loop), what would be the
magnetic force on the loop now?
(e) In the following diagram, two fixed L-shaped wires, separated by a distance x, are connected
by a wire that’s free to slide vertically.
The mass of the sliding wire, S, is m. If the sliding wire S crosses a region that contains a
uniform magnetic field B, how much current must be carried by the wire to keep S from
sliding down (due to its weight)?
3. The figure below shows two long, straight wires connected by a circular arc of radius x that
subtends a central angle ?. The current in the wire is I.
(a) Find the magnetic field (magnitude and direction) created at Point C. Write your answer in
terms of x, ?, I, and fundamental constants.
(b) A particle of charge +q is placed at Point C and released. Find the magnetic force on the
particle.
(c) A second long, straight wire is set up perpendicular to the plane of the page through C,
carrying the same current, I (directed out of the page), as the wire pictured in the diagram.
Determine the magnetic force per unit length between the wires.
4. For a conducting rod that carries a current I, the current density is defined as the current per
unit area: J = I/A.
Part 1. A homogeneous cylindrical rod of radius R carries a current whose current density, J,
is uniform (constant); that is, J does not vary with the radial distance, r, from the center of the
rod.
(a) Determine the total current, I, in the rod.
(b) Calculate the magnitude of the magnetic field for
(i) r < R
(ii) r > R, writing your answers in terms of r, R, I, and fundamental constants
Part 2. A nonhomogeneous cylindrical rod of radius R carries a current whose current
density, J, varies with the radial distance, r, from the center of the rod according to the
equation J = s r, where s is a constant.
(c) What are the units of s ?
(d) Determine the total current, I, in the rod.
(e) Calculate the magnitude of the magnetic field for
(i) r < R
(ii) r > R, writing your answers in terms of r, R, and I
Section II: Free Response
1. (a) The acceleration of an ion of charge q is equal to F
E
/m. The electric force is equal to qE,
where E = V/d. Therefore, a = qV/(dm).
(b) Using a = qV/(dm) and the equation , we get
As an alternate solution, notice that the change in the electrical potential energy of the ion
from the source S to the entrance to the magnetic-field region is equal to qV; this is equal
to the gain in the particle’s kinetic energy.
Therefore,
(c) (i) and (ii) Use the right-hand rule. Since v points to the right and B is into the plane of
the page, the direction of v × B is upward. Therefore, the magnetic force on a positively
charged particle (cation) will be upward, and the magnetic force on a negatively charged
particle (anion) will be downward. The magnetic force provides the centripetal force that
causes the ion to travel in a circular path. Therefore, a cation would follow Path 1 and an
anion would follow Path 2.
(d) Since the magnetic force on the ion provides the centripetal force,
Now, by the result of part (b),
Page 5


Section II: Free Response
1. The diagram below shows a simple mass spectrograph. It consists of a source of ions (charged
atoms) that are accelerated (essentially from rest) by the voltage V and enter a region
containing a uniform magnetic field, B. The polarity of V may be reversed so that both
positively charged ions (cations) and negatively charged ions (anions) can be accelerated.
Once the ions enter the magnetic field, they follow a semicircular path and strike the front wall
of the spectrograph, on which photographic plates are constructed to record the impact.
(a) What is the acceleration of an ion of charge q just before it enters the magnetic field?
(b) Find the speed with which an ion of charge q enters the magnetic field.
(c) (i) Which semicircular path, 1 or 2, would a cation follow?
(ii) Which semicircular path, 1 or 2, would an anion follow?
(d) Determine the mass of a cation entering the apparatus in terms of y, q, B, and V.
(d) Once a cation of charge q enters the magnetic field, how long does it take to strike the
photographic plate?
(f) What is the work done by the magnetic force in the spectrograph on a cation of charge q?
2. A wire of diameter d and resistivity ? is bent into a rectangular loop (of side lengths a and b)
and fitted with a small battery that provides a voltage V. The loop is placed at a distance cfrom
a very long, straight wire that carries a current I in the direction indicated in the diagram.
(Express all answers in terms of a, b, c, d, ?, V, I, m, B, x, and fundamental constants.)
(a) When the switch S is closed, find the current in the rectangular loop.
(b) What is the magnetic force (magnitude and direction) exerted on the loop by the long,
straight wire?
(c) The wire of the rectangular loop is then reshaped into a circle. What will be the radius of the
circular loop?
(d) If the loop constructed in part (c) were then threaded around the long, straight wire (so that
the straight wire passed through the center of the circular loop), what would be the
magnetic force on the loop now?
(e) In the following diagram, two fixed L-shaped wires, separated by a distance x, are connected
by a wire that’s free to slide vertically.
The mass of the sliding wire, S, is m. If the sliding wire S crosses a region that contains a
uniform magnetic field B, how much current must be carried by the wire to keep S from
sliding down (due to its weight)?
3. The figure below shows two long, straight wires connected by a circular arc of radius x that
subtends a central angle ?. The current in the wire is I.
(a) Find the magnetic field (magnitude and direction) created at Point C. Write your answer in
terms of x, ?, I, and fundamental constants.
(b) A particle of charge +q is placed at Point C and released. Find the magnetic force on the
particle.
(c) A second long, straight wire is set up perpendicular to the plane of the page through C,
carrying the same current, I (directed out of the page), as the wire pictured in the diagram.
Determine the magnetic force per unit length between the wires.
4. For a conducting rod that carries a current I, the current density is defined as the current per
unit area: J = I/A.
Part 1. A homogeneous cylindrical rod of radius R carries a current whose current density, J,
is uniform (constant); that is, J does not vary with the radial distance, r, from the center of the
rod.
(a) Determine the total current, I, in the rod.
(b) Calculate the magnitude of the magnetic field for
(i) r < R
(ii) r > R, writing your answers in terms of r, R, I, and fundamental constants
Part 2. A nonhomogeneous cylindrical rod of radius R carries a current whose current
density, J, varies with the radial distance, r, from the center of the rod according to the
equation J = s r, where s is a constant.
(c) What are the units of s ?
(d) Determine the total current, I, in the rod.
(e) Calculate the magnitude of the magnetic field for
(i) r < R
(ii) r > R, writing your answers in terms of r, R, and I
Section II: Free Response
1. (a) The acceleration of an ion of charge q is equal to F
E
/m. The electric force is equal to qE,
where E = V/d. Therefore, a = qV/(dm).
(b) Using a = qV/(dm) and the equation , we get
As an alternate solution, notice that the change in the electrical potential energy of the ion
from the source S to the entrance to the magnetic-field region is equal to qV; this is equal
to the gain in the particle’s kinetic energy.
Therefore,
(c) (i) and (ii) Use the right-hand rule. Since v points to the right and B is into the plane of
the page, the direction of v × B is upward. Therefore, the magnetic force on a positively
charged particle (cation) will be upward, and the magnetic force on a negatively charged
particle (anion) will be downward. The magnetic force provides the centripetal force that
causes the ion to travel in a circular path. Therefore, a cation would follow Path 1 and an
anion would follow Path 2.
(d) Since the magnetic force on the ion provides the centripetal force,
Now, by the result of part (b),
(e) Since the magnetic force cannot change the speed of a charged particle, the time required
for the ion to hit the photographic plate is equal to the distance traveled (the length of the
semicircle) divided by the speed computed in part (b):
(f) Since the magnetic force F
B
 is always perpendicular to a charged particle’s velocity
vector v, it can do no work on the particle. Thus, the answer is zero.
2. (a) The current in the rectangular loop is equal to V divided by the resistance of the
rectangular loop. Using the equation R = ? /A, we have
Therefore, the current in the rectangular loop is
(b) We use the equation F
B
 = I( × B) to find the magnetic force on each side of the
rectangular loop. By symmetry, the magnetic forces on the sides of length a have the same
magnitude but opposite direction, so they cancel. The total magnetic force on the loop is
equal to the sum of the magnetic forces on the sides of length b. Current in the rectangle
is directed clockwise, so the current in the bottom wire of length b (the one closer to the
wire) is directed to the left and the current in the top wire of length b is directed to the
right. Currents that are parallel feel an attractive force, while currents that are antiparallel
feel a repulsive force. Since the strength of the magnetic field at a distance r from a long,
straight wire carrying a current I is given by the equation B = (µ
0
/2p)(I/r), we find that
the magnetic force on the bottom wire is:
, upward (away from the long wire),
and the magnetic force on the top wire is:
, downward (toward the long wire).
Since F
B1
 has a greater magnitude than F
B2
, and the forces point in opposite directions
(with the direction of the net force equaling the direction of F
B1
), the magnitude of the
total magnetic force on the loop is equal to the difference between the magnitudes
of F
B1
 and F
B2
. Therefore,