Magnetic Fields and Magnetic Forces — AP Physics C: E&M
1. Core Concepts of Magnetic Fields ★★☆☆☆ ⏱ 3 min
A magnetic field ($\vec{B}$) is a vector field that only exerts force on moving electric charge. Unlike electric fields, magnetic forces never do work on charge because the force is always perpendicular to the charge's displacement. The SI unit for magnetic field is the tesla (T), where $1 \text{ T} = 1 \text{ N}/(\text{A·m})$.
Magnetic forces are tested in three key contexts on the AP exam: force on a single moving point charge, force on a bulk current-carrying wire, and torque on a current-carrying loop. This topic is also the foundation for mass spectrometry problems involving charged particle trajectories.
2. The Lorentz Force Law ★★☆☆☆ ⏱ 3 min
If no electric field is present, the law simplifies to $\vec{F} = q\vec{v} \times \vec{B}$. The magnitude of the magnetic force is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
Direction follows the right-hand rule for cross products: for positive $q$, point the fingers of your right hand along $\vec{v}$, curl them toward $\vec{B}$, and your thumb points to the force direction. For negative $q$, the force is exactly opposite this direction. A key property: magnetic force is always perpendicular to velocity, so it changes only direction of motion, not speed, so it never does work.
Exam tip: Always explicitly check the sign of the charge before writing your final direction. AP MCQs almost always include an option that matches the magnitude but has the wrong direction for negative charges to catch this common mistake.
3. Force on Current-Carrying Wires ★★★☆☆ ⏱ 3 min
A current in a wire is a continuous stream of moving charges, so the force on a wire can be derived directly from the Lorentz force law. For an infinitesimal segment of wire of length $d\vec{l}$, where $d\vec{l}$ points in the direction of conventional current, the force on the segment is $d\vec{F} = I d\vec{l} \times \vec{B}$, where $I$ is the current.
For a uniform magnetic field, integrating over the full length of the wire gives:
\vec{F} = I \left(\int d\vec{l}\right) \times \vec{B}
For a straight wire of total length $L$, this simplifies to $\vec{F} = I \vec{L} \times \vec{B}$, with magnitude $F = ILB\sin\theta$, where $\theta$ is the angle between $\vec{L}$ (current direction) and $\vec{B}$. A key result: the net force on any closed current loop in a uniform magnetic field is zero, because the integral of $d\vec{l}$ around a closed loop equals zero.
4. Motion of Charged Particles in Uniform Magnetic Fields ★★★☆☆ ⏱ 3 min
When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force that maintains uniform circular motion. Equating the Lorentz force magnitude to centripetal force gives:
qvB = \frac{mv^2}{r}
Rearranging gives the radius of the circular path:
r = \frac{mv}{qB} = \frac{p}{qB}
where $p = mv$ is the particle's momentum. The cyclotron angular frequency is $\omega = \frac{qB}{m}$, which is independent of the particle's speed or radius, the property that makes cyclotron particle accelerators work. If the particle has a velocity component parallel to $\vec{B}$, that component experiences no force, so the particle follows a helical path. This behavior is the basis of mass spectrometry, where particles of different mass are separated by their radius of curvature.
Exam tip: AP problems intentionally ask for diameter (the quantity measured experimentally in mass spectrometers) much more often than radius. Always confirm what quantity you are asked for before writing your final answer.
5. Torque on Current-Carrying Loops ★★★★☆ ⏱ 2 min
Even though the net force on a closed current loop in a uniform magnetic field is zero, there is a net torque that rotates the loop to align its magnetic dipole moment with the magnetic field. The torque formula is:
\vec{\tau} = \vec{\mu} \times \vec{B}
where $\vec{\mu}$, the magnetic dipole moment of the loop, has magnitude $\mu = IA$, with $I$ the current and $A$ the area of the loop. The direction of $\vec{\mu}$ is along the normal to the plane of the loop, found via right-hand rule: curl your fingers along the direction of current, and your thumb points to $\vec{\mu}$. The magnitude of torque is $\tau = IAB\sin\theta$, where $\theta$ is the angle between $\vec{\mu}$ and $\vec{B}$. The potential energy of the dipole in the B field is $U = -\vec{\mu} \cdot \vec{B}$, so the lowest energy (equilibrium) state is when $\vec{\mu}$ is aligned with $\vec{B}$. This relationship is the operating principle of electric motors and galvanometers.
Exam tip: If the problem gives the angle between the plane of the loop and B, always subtract from 90 degrees to get the correct $\theta$ for the torque formula.
6. Concept Check
Common Pitfalls
Why: Students memorize the right-hand rule for positive charge and do not account for the sign of $q$ in the cross product.
Why: Students default to outputting radius from the common formula, and do not read the question’s request carefully.
Why: Students confuse magnetic force with other constant forces and forget its key direction property.
Why: Problems often describe orientation relative to the plane, so students mix up plane angle and normal angle.
Why: Students memorize the zero net force rule and forget it only applies to uniform B.