AP Physics 2: Force on Moving Charges in B Fields

Understanding how magnetic fields affect moving charged particles is fundamental to technologies from cathode-ray tubes to MRI machines and particle accelerators. This concept, governed by the Lorentz force law, explains why charged particles curve in magnetic fields and forms the basis for devices that filter and analyze particles by their speed or mass. Mastering this topic requires a solid grasp of vector mathematics and its application to physical motion.

The Lorentz Force Law: F = qv × B

The magnetic force on a single moving charge is given by the vector equation ${\vec{F}}_B=q\vec{v}\times \vec{B}$. This is the magnetic component of the full Lorentz force law. Here, $q$ is the charge of the particle (including its sign, in Coulombs), $\vec{v}$ is its instantaneous velocity vector, and $\vec{B}$ is the magnetic field vector. The "×" symbol represents the cross product, a specific mathematical operation between two vectors that produces a third vector perpendicular to the plane containing the first two.

The cross product dictates both the magnitude and direction of the force. The magnitude is calculated as $F_B=|q|vB\sin \theta$, where $\theta$ is the angle between the vectors $\vec{v}$ and $\vec{B}$. This formula reveals two critical conditions: if the charge is stationary ($v=0$), the magnetic force is zero, and if the velocity is parallel or anti-parallel to the field ($\theta =0^{\circ }$ or $180^{\circ }$), the force is also zero because $\sin 0^{\circ }=0$. The maximum force occurs when the velocity is perpendicular to the field ($\theta =90^{\circ }$), simplifying the magnitude to $F_{B,\text{max}}=|q|vB$.

The direction of the force is given by the right-hand rule for cross products. For a positive charge, point your fingers in the direction of $\vec{v}$, curl them toward $\vec{B}$, and your thumb points in the direction of ${\vec{F}}_B$. For a negative charge (like an electron), the force direction is reversed. Crucially, because the force vector is perpendicular to the velocity vector, magnetic forces do no work on a charged particle. They can change the particle's direction but not its kinetic energy or speed.

Circular Motion in a Uniform Magnetic Field

When a charged particle enters a uniform magnetic field with a velocity perfectly perpendicular to the field lines, the resulting force is always perpendicular to its motion. This is the condition for uniform circular motion. The magnetic force acts as the centripetal force, constantly pulling the particle into a circular path.

We can derive the characteristics of this motion by setting the magnetic force equal to the centripetal force: $$F_B=qvB=\frac{m{v}^2}{r}$$ Solving for the radius of the circular path, or cyclotron radius, gives: $$r=\frac{mv}{qB}$$ This equation tells us the radius is directly proportional to the particle's momentum ($mv$) and inversely proportional to its charge and the field strength. A more massive or faster particle will curve less (larger radius), while a stronger field or larger charge causes a tighter curve (smaller radius).

The time it takes for the particle to complete one full circle is the period ($T$). Since distance is circumference ($2\pi r$) and speed is constant: $$T=\frac{2\pi r}{v}=\frac{2\pi m}{qB}$$ Notice that the period is independent of the particle's speed and the radius of its path! It depends only on the mass, charge, and magnetic field strength. The corresponding frequency, $f=1/T=\frac{qB}{2\pi m}$, is called the cyclotron frequency.

If the particle's velocity has a component parallel to the magnetic field, that component is unaffected (since $F_B=0$ for parallel motion). The perpendicular component causes circular motion. The superposition of this circular motion and the constant parallel velocity results in a helical path that spirals along the field line.

Applications: Velocity Selectors and Mass Spectrometers

The principles of motion in combined electric and magnetic fields lead to invaluable practical devices. A velocity selector uses perpendicular electric and magnetic fields to filter particles by speed. Particles enter a region where a uniform electric field $\vec{E}$ and magnetic field $\vec{B}$ are oriented perpendicular to each other and to the initial particle beam.

The electric force on a charge is ${\vec{F}}_E=q\vec{E}$. The magnetic force is ${\vec{F}}_B=q\vec{v}\times \vec{B}$. If the fields are arranged such that these forces are in opposite directions, only particles with the exact speed where the forces cancel ($F_E=F_B$) will travel in a straight line and pass through the selector. Setting the magnitudes equal: $$qE=qvB\Rightarrow v=\frac{E}{B}$$ Particles moving slower or faster than this speed will be deflected upwards or downwards and hit the walls of the apparatus. This provides a pure beam of particles all moving at the same known speed.

A mass spectrometer often follows a velocity selector. After particles are selected for a specific speed $v$, they enter a region of uniform magnetic field $\overrightarrow{B^{\prime }}$ perpendicular to their velocity. They undergo circular motion with radius $r=\frac{mv}{q{B}^{\prime }}$. Since $v$, $q$, and $B^{\prime }$ are now known, measuring the radius of curvature $r$ directly reveals the mass-to-charge ratio ($m/q$) of the particle: $$\frac{m}{q}=\frac{r{B}^{\prime }}{v}=\frac{r{B}^{\prime }B}{E}$$ By comparing radii, scientists can identify different isotopes (which have the same charge but different masses) present in a sample, a technique foundational to chemistry and nuclear physics.

Common Pitfalls

1. Forgetting the Absolute Value in the Magnitude Formula: The magnitude formula is $F_B=|q|vB\sin \theta$. The charge's sign ($+$ or $–$) does not affect the magnitude of the force, only its direction. Using $q$ instead of $|q|$ can give a negative magnitude, which is nonsensical.
2. Misapplying the Right-Hand Rule for Negative Charges: The standard right-hand rule gives the force direction for a positive charge. For a negative charge like an electron, you have two options: use the left hand, or use the right-hand rule and then reverse the resulting direction. Consistently applying one method is key.
3. Assuming Magnetic Fields Can Do Work: A common conceptual error is to think a magnetic force can speed up or slow down a charged particle. Because ${\vec{F}}_B$ is always perpendicular to $\vec{v}$, the work done is $W=\vec{F}\cdot \vec{d}=0$. It changes direction, not speed. Any change in a particle's kinetic energy in a real device (like a cyclotron) is due to electric fields, not magnetic fields.
4. Confusing the Roles of E and B in a Velocity Selector: In the velocity selector equation $v=E/B$, it is critical that the fields are perpendicular. Students sometimes mistakenly try to apply this condition when the fields are not arranged to produce opposing forces. Always draw a diagram to confirm the force vectors from each field are colinear and opposite.

Summary

• The magnetic force on a moving charge is ${\vec{F}}_B=q\vec{v}\times \vec{B}$. Its magnitude is $F_B=|q|vB\sin \theta$, and its direction is perpendicular to the plane of $\vec{v}$ and $\vec{B}$, given by the right-hand rule (reversed for negative charges).
• In a uniform B-field, a velocity perpendicular to the field results in uniform circular motion. The radius is $r=\frac{mv}{qB}$ and the period is $T=\frac{2\pi m}{qB}$, which is independent of speed.
• A velocity selector uses crossed E and B fields where forces balance ($v=E/B$) to allow only particles with a specific speed to pass through undeflected.
• A mass spectrometer uses a velocity selector followed by a magnetic field to cause circular motion, allowing the measurement of mass-to-charge ratios via the radius of curvature: $\frac{m}{q}=\frac{r{B}^{\prime }B}{E}$.
• Magnetic forces are centripetal and do no work; they change a particle's direction but not its kinetic energy or speed.