How Do You Determine the Charge-to-Mass Ratio of a Particle From Its Path in a Magnetic Field?

For both SL and HL: To determine the magnitude of a particle's charge-to-mass ratio, measure the circular-path radius $r$, determine its speed $v$, and use the uniform magnetic field strength $B$. When velocity is perpendicular to the field, use

$$\frac{|q|}{m}=\frac{v}{Br}.$$

The Reasoning

A particle of charge $q$ moving perpendicular to a uniform magnetic field experiences a magnetic force of magnitude $F_B=|q|vB$. Since this force is always perpendicular to the velocity, it changes direction but not speed.

The magnetic force therefore supplies the centripetal force required for circular motion:

$$|q|vB=\frac{mv^2}{r}.$$

Cancel $v$ and rearrange:

$$\frac{|q|}{m}=\frac{v}{Br}.$$

The result gives magnitude only; the sign must be found from the curvature.

For example, a particle moving at $3.0\times10^6\ \text{m s}^{-1}$ follows a path of radius $0.20\ \text{m}$ in a $0.50\ \text{T}$ field. Substitution gives:

$$\frac{|q|}{m}=\frac{3.0\times10^6}{(0.50)(0.20)}=3.0\times10^7\ \text{C kg}^{-1}.$$

A larger radius at fixed $v$ and $B$ means a smaller (|q|/m).

The radius alone is insufficient because the ratio also depends on speed. Common ways to obtain $v$ are:

For the last case, electrical potential energy becomes kinetic energy: $|q|V=\frac{1}{2}mv^2$. Combining this with $\mathrm{\mid }q\mathrm{\mid }vB$ removes and produces the table's expression.

To identify the sign, use the direction of $\boldsymbol{v}\times\boldsymbol{B}$, which gives the force direction for a positive charge. A negative particle curves oppositely.

A common misconception is that the magnetic field changes the particle's speed. It does no work because its force is perpendicular to the instantaneous displacement; only the velocity's direction changes.

Exam Technique

In an IB Physics D.3 response, state that magnetic force provides centripetal force, equate the forces, substitute consistently, and show the rearrangement. For direction questions, apply the hand rule to a positive charge first, then reverse the result for a negative charge.