12.2 Magnetism and Moving Charges

A moving charge creates a magnetic field, and an external magnetic field exerts a force on a charge moving through it. The magnetic force has magnitude $F_B = qvB\sin\theta$, points perpendicular to both velocity and field, and follows the right-hand rule for direction.

Why This Matters for the AP Physics 2 Exam

Magnetism and moving charges is part of Unit 12, which carries about 12 to 15 percent of the exam. This topic shows up in both multiple-choice and free-response questions, where you may need to predict how the magnetic force changes when you adjust charge, speed, field strength, or angle. On free-response items, you have to explain your reasoning step by step. Saying "the force points right because of the right-hand rule" does not support a stronger score. You need to connect the equation or principle to a clear explanation of why the force points where it does.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Key Takeaways

• A single moving charge produces a magnetic field that is stronger at higher speeds, weaker farther away, and depends on the angle between the velocity and the position vector to the point.
• The magnetic force on a moving charge is $F_B = qvB\sin\theta$, so it depends on charge, speed, field strength, and angle.
• The magnetic force is always perpendicular to both the velocity and the magnetic field, and its direction comes from the right-hand rule (reverse it for negative charges).
• Force is zero when velocity is parallel to the field ($\theta = 0^\circ$ or $180^\circ$) and maximum when velocity is perpendicular ($\theta = 90^\circ$).
• When both electric and magnetic fields are present, the charge feels independent forces that you add as vectors.
• The Hall effect is the potential difference created across a conductor when a magnetic field pushes its moving charges sideways.

Magnetic Field Produced by Moving Charges

A charged particle that moves through space creates a magnetic field around it. This is one of the core links between electricity and magnetism.

For AP Physics 2, describe this relationship qualitatively. The magnetic field from a single moving charge:

• Gets stronger when the charge moves faster.
• Gets weaker farther from the charge.
• Depends on the angle between the velocity vector and the position vector pointing from the charge to your point of interest.

The field is zero when those two vectors are parallel and greatest when they are perpendicular. At any chosen point, the field direction is perpendicular to both the velocity and the position vector, and you find that direction with the right-hand rule for a positive charge. For a negative charge, the direction is reversed.

Force Exerted on Moving Charges by Magnetic Fields

Magnetic Interactions Between Moving Charges

Moving charges interact with each other through magnetic forces. This interaction is behind many electromagnetic phenomena.

• When two charges move, each creates a magnetic field that exerts a force on the other.
• These magnetic interactions depend on the motion of the charges.
• Unlike electric forces, which act along the line connecting two charges, magnetic forces act perpendicular to the motion.

Force on a Charge Moving Through a Field

When a charged particle moves through a magnetic field, it feels a force that depends on several quantities. The magnitude is:

$F_B = qvB\sin\theta$

Where:

• $F_B$ is the magnetic force
• $q$ is the charge of the particle
• $v$ is the speed of the particle
• $B$ is the magnetic field strength
• $\theta$ is the angle between the velocity and magnetic field vectors

The direction of this force is always perpendicular to both the magnetic field and the charge's velocity.

Use the right-hand rule to find the direction:

1. Point your fingers in the direction of the velocity $v$.
2. Curl them toward the magnetic field $B$.
3. Your thumb points in the direction of the force $F$ for a positive charge.

For a negative charge, the force points opposite your thumb.

When a charge moves parallel to the field ($\theta = 0^\circ$ or $180^\circ$), it feels no magnetic force. The maximum force happens when the charge moves perpendicular to the field ($\theta = 90^\circ$).

Independent Forces from Electric and Magnetic Fields

In a region with both an electric field and a magnetic field, a moving charge feels two separate forces:

• The electric force: $F_E = qE$ (along the electric field for a positive charge)
• The magnetic force: $F_B = qvB\sin\theta$ (perpendicular to both velocity and field)

These forces act independently. Calculate each one separately, then add them as vectors to get the net force on the charge.

Hall Effect

The Hall effect happens when charges moving through a conductor meet an external magnetic field that has a component perpendicular to their motion. The magnetic force pushes the moving charges sideways, so charge piles up on one side of the conductor and the opposite side ends up with the opposite charge. This separation creates a potential difference across the conductor, called the Hall voltage. For AP Physics 2, describe this qualitatively as the result of the magnetic force acting on moving charges in the conductor.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify what you are given: charge, speed, field strength, and angle. Plug into $F_B = qvB\sin\theta$.
• For magnitude problems with an electron or other negative charge, use the absolute value of the charge, then handle direction separately.
• Watch the angle. Quantitative force questions stick to $0^\circ$, $90^\circ$, and $180^\circ$, so $\sin\theta$ is usually 0 or 1.

Free Response

• Do not stop at "right-hand rule." Walk through the steps: where your fingers point, how they curl, where your thumb ends up, and whether you flip it for a negative charge.
• When both fields are present, state clearly that the electric and magnetic forces are independent and show the vector addition.
• Tie any prediction about increasing or decreasing force back to the variable you changed in the equation.

Common Trap

• The magnetic force never speeds up or slows down a charge, because it is always perpendicular to the velocity. It changes direction, not speed.

Practice Problem 1: Magnetic Force on a Moving Charge

Solution

Use the magnetic force equation:

$F_B = qvB\sin\theta$

Since you want the magnitude, use the magnitude of the charge: $|q| = 1.6 × 10^{-19}$ C. Because the velocity is perpendicular to the field, $\sin 90^\circ = 1$.

$F_B = |q|vB\sin\theta$ $F_B = (1.6 × 10^{-19} \text{ C})(2.0 × 10^6 \text{ m/s})(0.50 \text{ T})(1)$ $F_B = 1.6 × 10^{-13} \text{ N}$

The magnitude of the force is 1.6 × 10^-13 N. To find the direction, use the right-hand rule for a positive charge and then reverse the result because the particle is an electron.

Practice Problem 2: Magnetic Force with Parallel Motion

Solution

Use the magnetic-force equation:

$F_B = qvB\sin\theta$

Because the proton moves parallel to the field, $\theta = 0^\circ$, so $\sin 0^\circ = 0$.

Therefore,

$F_B = qvB(0) = 0$

The magnetic force is $0\ \text{N}$.

Common Misconceptions

• "A magnetic field always pushes on a charge." It only pushes on a charge that is moving and that has a velocity component perpendicular to the field. A still charge or a charge moving straight along the field feels no magnetic force.
• "The magnetic force does work on the charge." Because the force is always perpendicular to the velocity, it does no work and cannot change the charge's speed. It only bends the path.
• "The right-hand rule works the same for electrons." For negative charges, the force points opposite to what your right hand gives you, so always flip the result.
• "Electric and magnetic forces combine into one formula." In a region with both fields, treat the forces separately and add them as vectors.
• "Bigger angle always means bigger force." The force depends on $\sin\theta$, so it peaks at $90^\circ$ and drops to zero at $0^\circ$ and $180^\circ$, not steadily with angle.

Related AP Physics 2 Guides

• 12.1 Magnetic Fields
• 12.4 Electromagnetic Induction and Faraday's Law
• 12.3 Magnetism and Current-Carrying Wires