Magnetism and Moving Charges

Introduction: Why moving charges matter ⚡🧲

students, you already know that electric forces come from electric charges. Magnetism is closely related, but it shows up in a special way: it is strongly connected to moving charges. That connection is a major idea in AP Physics 2 and is part of the broader topic of Magnetism and Electromagnetism. In this lesson, you will learn how magnetic fields act on moving charges, how to determine the direction of the force, and why this idea is used in technology such as motors, speakers, and particle accelerators.

Learning objectives

By the end of this lesson, you should be able to:

• Explain the main ideas and vocabulary connected to magnetism and moving charges.
• Use algebraic reasoning to find the magnetic force on a charge.
• Determine the direction of magnetic force using right-hand rules.
• Connect magnetic force on moving charges to larger electromagnetic systems.
• Use examples and evidence to explain why moving charges behave differently in magnetic fields.

A key idea to remember is this: a magnetic field does not push on a charge unless the charge is moving. That makes magnetism different from electric fields, which can push on charges whether they are moving or not.

Magnetic fields and what they do

A magnetic field is a region where magnetic effects can be detected. It is represented by the symbol $\mathbf{B}$, and its unit is the tesla, written as $\mathrm{T}$. Magnetic fields are often shown with field lines, just like electric fields, but the meaning is a little different.

Magnetic field lines indicate the direction a north pole would move if it were free to move. Around a bar magnet, the lines leave the north pole and enter the south pole outside the magnet. Near a current-carrying wire, the magnetic field forms circles around the wire. This is important because electric current is moving charge.

For AP Physics 2, you do not need to memorize every possible magnetic field pattern at once, but you should know the core idea: magnetic fields are created by moving charges and currents, and they act on moving charges.

Real-world example: the magnetic strip on a credit card is not the main focus here, but a loudspeaker is a great example. Inside a speaker, an electric current moves through a coil in a magnetic field. The force on that moving charge causes the coil to move, which pushes air and creates sound 🔊.

The magnetic force on a moving charge

The most important equation in this lesson is the magnetic force on a charged particle:

$$F = qvB\sin\theta$$

Here:

• $F$ is the magnitude of the magnetic force,
• $q$ is the charge,
• $v$ is the speed of the particle,
• $B$ is the magnetic field strength,
• $\theta$ is the angle between the particle’s velocity and the magnetic field.

This equation tells us several important things:

1. If $v = 0$, then $F = 0$. A stationary charge feels no magnetic force.
2. If the particle moves parallel to the magnetic field, then $\theta = 0^\circ$ or $180^\circ$, so $\sin\theta = 0$ and the force is zero.
3. If the particle moves perpendicular to the field, then $\theta = 90^\circ$, so $\sin\theta = 1$ and the force is maximum.
4. The force depends on the amount of charge. A larger $|q|$ gives a larger force.

Notice something very important: the magnetic force depends on the direction of motion, not just on position. That means magnetism behaves differently from many everyday forces.

Example 1: force size

Suppose a proton with charge $q = 1.60\times10^{-19}\,\mathrm{C}$ moves at speed $v = 2.0\times10^6\,\mathrm{m/s}$ through a magnetic field of $B = 0.50\,\mathrm{T}$, perpendicular to the field.

Since $\theta = 90^\circ$, we use:

$$F = qvB$$

$$F = (1.60\times10^{-19})(2.0\times10^6)(0.50)$$

$$F = 1.6\times10^{-13}\,\mathrm{N}$$

That is a very small force, but it can still have a major effect on tiny particles moving very fast.

Direction of the magnetic force

The magnetic force is a vector, so direction matters. The direction is found using the right-hand rule. For a positive charge, point your fingers in the direction of the velocity $\mathbf{v}$, curl them toward the magnetic field $\mathbf{B}$, and your thumb points in the direction of the force $\mathbf{F}$.

For a negative charge, the force points in the opposite direction.

This is a common place where students make mistakes, so students, take your time and always check the sign of the charge.

Example 2: direction

Imagine a positive charge moving to the right while the magnetic field points into the page. Using the right-hand rule, the force points upward. If the charge were negative instead, the force would point downward.

This direction is not random. The force is always perpendicular to both the velocity and the magnetic field. That is why the force changes the direction of motion rather than the speed, at least when the speed stays constant.

Why magnetic force changes direction, not speed

A magnetic force is always perpendicular to the particle’s velocity when the particle moves through the field. Because of this, the force does no work on the particle.

Work is related to force and displacement by:

$$W = Fd\cos\theta$$

If the force is perpendicular to the motion, then $\theta = 90^\circ$ and $\cos\theta = 0$, so:

$$W = 0$$

Since no work is done, the particle’s kinetic energy does not change. That means the speed stays the same, even though the velocity changes because the direction changes.

This is why a charged particle in a magnetic field can move in a circle or spiral. The force acts like a centripetal force, pulling the particle toward the center of its curved path.

Circular motion connection

For uniform circular motion, the centripetal force is:

$$F_c = \frac{mv^2}{r}$$

If a charged particle moves perpendicular to a magnetic field, then the magnetic force provides the centripetal force:

$$qvB = \frac{mv^2}{r}$$

Solving for the radius gives:

$$r = \frac{mv}{qB}$$

This equation shows that faster particles or particles with larger mass travel in larger circles, while stronger magnetic fields or larger charges make the circle smaller.

This idea is used in devices like mass spectrometers, which separate particles based on charge and mass. It also helps in particle physics and medical equipment.

Magnetic force on a current-carrying wire

Since electric current is the movement of charge, a wire carrying current also experiences magnetic force in a magnetic field. The equation is:

$$F = ILB\sin\theta$$

Here:

• $I$ is current,
• $L$ is the length of wire in the field,
• $B$ is the magnetic field,
• $\theta$ is the angle between the wire and the field.

This is really the same idea as the force on moving charges, but applied to many charges moving together in a wire.

This principle is the basis of an electric motor. In a motor, current flows through a coil inside a magnetic field. The forces on different parts of the coil create a turning effect, called torque, which makes the coil rotate. That is how electrical energy becomes mechanical motion.

Example 3: motor connection

If one side of a coil experiences an upward force and the opposite side experiences a downward force, the coil starts to spin. This is not because of gravity or friction, but because the magnetic field pushes on the moving charges in opposite directions on different sides of the coil.

That same basic physics is also behind galvanometers and some types of meters that detect current.

Big-picture connection to electromagnetism

students, magnetism and moving charges are not an isolated topic. They are part of electromagnetism, where electric and magnetic effects are connected.

Here is the big picture:

• Electric charges create electric fields.
• Moving charges and currents create magnetic fields.
• Magnetic fields act on moving charges.
• Changing magnetic fields can produce electric fields, which is the foundation of electromagnetic induction.

In this lesson, the focus is mainly on how magnetic fields affect moving charges. But this is only one part of the larger story. Together, electric fields, magnetic fields, and electromagnetic induction explain many technologies in the modern world.

Examples include:

• electric motors,
• speakers,
• magnetic resonance imaging in medicine,
• particle accelerators,
• generators.

These devices all rely on the relationship between charge, motion, and magnetic fields.

Conclusion

Magnetism and moving charges are connected by a few powerful ideas. A magnetic field, written as $\mathbf{B}$, exerts a force on a charge only when that charge is moving. The magnetic force is given by $F = qvB\sin\theta$, and its direction is always perpendicular to the velocity and the magnetic field. Because of that perpendicular force, magnetic fields change the direction of motion without changing speed. This leads to circular or curved paths and explains how many devices work in the real world. Understanding these ideas gives you a strong foundation for the rest of magnetism and electromagnetism in AP Physics 2.

Study Notes

• A magnetic field is represented by $\mathbf{B}$ and measured in teslas, $\mathrm{T}$.
• Magnetic force acts on moving charges, not stationary charges.
• The force on a moving charge is $F = qvB\sin\theta$.
• The force is maximum when $\theta = 90^\circ$ and zero when the motion is parallel to the field.
• The right-hand rule gives the force direction for a positive charge; reverse it for a negative charge.
• Magnetic force is perpendicular to velocity, so it changes direction but not speed.
• When $v\perp B$, the particle can move in a circle with radius $r = \frac{mv}{qB}$.
• A current-carrying wire in a magnetic field experiences force $F = ILB\sin\theta$.
• Electric motors use magnetic force on current to produce rotation.
• Magnetism is a major part of electromagnetism, which also includes electric fields and electromagnetic induction.