How Do You Calculate the Force on a Moving Charge in a Magnetic Field (F = qvB sin θ)?

Use $F=qvB\sin\theta$, where $q$ is the magnitude of the charge, $v$ is its speed, $B$ is the magnetic field strength, and $\theta$ is the angle between the velocity and magnetic field. This equation is part of IB Physics D.3 for both SL and HL.

The Reasoning and Mechanism

The magnetic force acts on a charge only when the charge is moving and has a velocity component perpendicular to the magnetic field. The factor $\sin\theta$ selects this perpendicular component:

$$v_{\perp}=v\sin\theta$$

Therefore, $F=qv_{\perp}B=qvB\sin\theta$. Use the magnitude (|q|) when calculating the force magnitude; the charge's sign affects only the force direction.

For example, a charge of magnitude $2.0\times10^{-6}\,\text{C}$ moves at $3.0\times10^4\,\text{m s}^{-1}$ through a field at :

$$F=(2.0\times10^{-6})(3.0\times10^4)(0.20)\sin30^\circ =6.0\times10^{-3}\,\text{N}$$

Determine direction using the appropriate hand rule for a positive charge. For a negative charge, reverse that direction. The force is perpendicular to both the velocity and field, so it does no work and changes the direction of motion rather than the speed.

In vector form, the relationship is $\vec F=q(\vec v\times\vec B)$. For a positive charge, point the fingers of your right hand along , curl them toward , and your thumb gives . For an electron, reverse the result. If the velocity is perpendicular to a uniform field, the force may provide the , producing circular motion: , so . Because the force is always perpendicular to the instantaneous velocity, ; kinetic energy and speed remain constant. A common misconception is that a magnetic field speeds up an isolated charge. It cannot; it bends the path.

IB Exam Technique

For a Calculate question, show the equation, substitution, angle evaluation, and final answer in newtons. A common misconception is using the angle to a diagram's surface rather than the angle directly between $\vec v$ and $\vec B$. Also avoid treating magnetic force as parallel to the field.