Magnetic Forces Do Not Work
Theorem1
Magnetic forces do not do work.
Explanation
In situations with magnetic forces, when particles or objects move, it might seem as though the magnetic forces are doing work. However, this is not the case.
Proof
Work is the product of force and displacement.
$$ W=\int \mathbf{F} \cdot d\mathbf{l} $$
The work done by the magnetic force is
$$ W_{\text{mag}}=\int \mathbf{F}_{\text{mag}} \cdot d\mathbf{l} = \int Q(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l} $$
Given $d\mathbf{l} = \dfrac{d\mathbf{l}}{dt}dt = \mathbf{v}dt$,
$$ W_{\text{mag}} = \int Q( \mathbf{v} \times \mathbf{B})\cdot \mathbf{v}dt $$
Here, $(\mathbf{v}\times \mathbf{B})$ is a vector perpendicular to $\mathbf{v}$. Since a vector perpendicular to $\mathbf{v}$ is dotted with $\mathbf{v}$, the result is naturally $0$. Therefore,
$$ W_{\text{mag}} =\int 0 dt=0 $$
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David J. Griffiths, Introduction to Electrodynamics (4th Edition, 2014), p232-233 ↩︎