Magnetic Source Problems
Level 5 Physics
January 2013
1. Suppose there are two parallel wires of length ` separated by distance a carrying currents I1
and I2 .
(a) If the currents travel in the same direction, do the wires move closer or farther apart?
(b) If the currents travel in the opposite direction, do the wires move closer or farther apart?
(c) What is the magnitude of the magnetic force that either of the wires experiences?
2. Consider an infinite length straight wire of radius R carrying a current I of uniform current
density. Let r be the distance from the central axis of the wire.
(a) Find the magnetic field where r < R.
(b) Find the magnetic field where r ≥ R.
(c) Create a graph with the magnitude of the magnetic field B on the y-axis with r on the
x-axis. Where is the magnitude of the magnetic field greatest?
Hint: Use the Amperian loops shown in the diagram below in conjunction with Ampere’s
Law.
3. A solenoid is a tightly wound coil of wire. The magnetic field lines for a sample solenoid
are shown below. For an ideal solenoid, which has infinite length and tightly packed coils,
the resulting magnetic field inside the solenoid becomes fairly uniform while the magnetic
field outside the solenoid vanishes. What is the magnitude of the magnetic field inside an
ideal solenoid carrying a current I with n coil turns per unit length? Does the magnitude
depend on the position inside the solenoid?
Hint: Use a rectangular Amperian loop as shown in the second diagram in conjunction
with Ampere’s Law.
Problems adapted from MIT 8.02 course notes
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4. A toroid is essentially a doughnut-shaped solenoid as seen below. What is the magnitude
of the magnetic field inside a toroid with current I, radius r, and N turns? Does the
magnitude depend on the position inside the toroid?
Hint: Use a circular Amperian loop as shown in the diagram in conjunction with Ampere’s
Law.
5. Consider an infinitely large plate of thickness b lying in the xy plane with a uniform current
density~J = J0 î. Let z be the displacement from the middle of the plate.
(a) Find the magnetic field above and below the plate. (z ≥
b
2
and z ≤ − 2b )
(b) Find the magnetic field inside the plate (− 2b < z < 2b ).
Hint: Use rectangular Amperian loops as shown in the second diagram in conjunction with
Ampere’s Law.
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