Magnetic Forces and Fields — Hard Practice Quiz

A Physics cheat sheet for Magnetic Forces and Fields — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. A proton enters a uniform magnetic field $B$ perpendicular to its velocity $v$. If the magnetic field strength is doubled, how does the magnetic force on the proton and the radius of its circular path change?

   • Force doubles, radius halves.
   • Force doubles, radius doubles.
   • Force halves, radius doubles.
   • Force halves, radius halves.

   Answer: Force doubles, radius halves.

2. Two long, parallel wires are separated by a distance $d$. The first wire carries current $I_1$ and the second wire carries current $I_2$. If the magnitude of the force per unit length on the second wire due to the first is $F/L$, derive an expression for the current $I_2$ in terms of $F/L$, $d$, $I_1$, and $\mu_0$.

   • $I_2 = \frac{2 \pi d (F/L)}{\mu_0 I_1}$
   • $I_2 = \frac{\mu_0 I_1 (F/L)}{2 \pi d}$
   • $I_2 = \frac{2 \pi d I_1}{\mu_0 (F/L)}$
   • $I_2 = \frac{\mu_0 (F/L)}{2 \pi d I_1}$

   Answer: $I_2 = \frac{2 \pi d (F/L)}{\mu_0 I_1}$

3. A square coil with $N$ turns and side length $s$ carries current $I$. It is placed inside a long solenoid with $n$ turns per unit length carrying current $I_s$. The coil's normal is perpendicular to the solenoid's axis. If the current in the solenoid $I_s$ is doubled, and the number of turns in the coil $N$ is halved, how does the maximum torque on the coil change?

   • Torque remains unchanged.
   • Torque doubles.
   • Torque halves.
   • Torque quadruples.

   Answer: Torque remains unchanged.

4. A proton moves with velocity $v$ directly towards a long straight wire carrying current $I$. The proton is initially at a distance $r$ from the wire. What is the direction of the magnetic force on the proton at the instant it is moving towards the wire?

   • Parallel to the wire, in the direction of current.
   • Parallel to the wire, opposite to the direction of current.
   • Perpendicular to both the wire and the proton's velocity, pointing away from the wire.
   • Perpendicular to both the wire and the proton's velocity, pointing towards the wire.

   Answer: Parallel to the wire, opposite to the direction of current.

5. A particle of mass $m$ and charge $q$ moves in a circular path of radius $r$ at the center of a circular loop of radius $R$ with $N$ turns carrying current $I$. The particle's velocity is perpendicular to the plane of the loop. Derive an expression for the particle's speed $v$ in terms of $m, q, r, R, N, I,$ and $\mu_0$.

   • $v = \frac{|q| r N \mu_0 I}{2 m R}$
   • $v = \frac{2 m R}{|q| r N \mu_0 I}$
   • $v = \frac{m R}{|q| r N \mu_0 I}$
   • $v = \frac{|q| r \mu_0 I}{2 m R N}$

   Answer: $v = \frac{|q| r N \mu_0 I}{2 m R}$

6. A straight wire of length $L$ carrying current $I_w$ is placed along the axis of a long solenoid. The solenoid has $n$ turns per unit length and carries current $I_s$. What is the magnetic force on the wire?

   • $F = \mu_0 n I_s I_w L$
   • $F = \frac{\mu_0 n I_s I_w L}{2}$
   • $F = 0$
   • $F = \mu_0 n I_s I_w L \sin(45^\circ)$

   Answer: $F = 0$

7. A charged particle enters a region with a uniform magnetic field $B$ and moves in a circular path. If the kinetic energy of the particle is quadrupled, how must the magnetic field strength $B$ change to maintain the same radius of the circular path?

   • $B$ must be doubled.
   • $B$ must be quadrupled.
   • $B$ must be halved.
   • $B$ must be quartered.

   Answer: $B$ must be doubled.

8. A rectangular coil of $N$ turns, length $L$, and width $w$ carries current $I_c$. It is placed such that its length $L$ is parallel to a long straight wire carrying current $I_w$. The nearest side of the coil is at a distance $r_1$ from the wire, and the farthest side is at a distance $r_2 = r_1 + w$. Assuming the coil's plane contains the wire, derive an expression for the magnitude of the net magnetic force on the coil.

   • $F_{net} = \frac{N \mu_0 I_w I_c L w}{2 \pi r_1 r_2}$
   • $F_{net} = \frac{\mu_0 I_w I_c L w}{2 \pi r_1 r_2}$
   • $F_{net} = \frac{N \mu_0 I_w I_c L}{2 \pi} \left(\frac{1}{r_1} + \frac{1}{r_2}\right)$
   • $F_{net} = \frac{\mu_0 I_w I_c L}{2 \pi} \left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

   Answer: $F_{net} = \frac{N \mu_0 I_w I_c L w}{2 \pi r_1 r_2}$

9. A proton is moving with speed $v$ along the axis of a long solenoid. The solenoid has $n$ turns per unit length and carries current $I$. What is the magnetic force on the proton?

   • $F = |q|v \mu_0 n I$
   • $F = \frac{|q|v \mu_0 n I}{2}$
   • $F = 0$
   • $F = |q|v \mu_0 n I \sin(45^\circ)$

   Answer: $F = 0$

10. A particle of charge $q$ and mass $m$ is moving with speed $v$ in a circular path of radius $r$ around a long straight wire carrying current $I$. The particle's velocity is perpendicular to the magnetic field produced by the wire. If the current $I$ in the wire is doubled, and the particle's speed $v$ is halved, how must the radius $r$ of its circular path change to maintain the same magnetic force on the particle?

    • Radius remains unchanged.
    • Radius doubles.
    • Radius halves.
    • Radius quadruples.

    Answer: Radius remains unchanged.