📂Electrodynamics

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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