Question

Why Magnetic Force Does No Work?

Answer 1

Explanation:

1. The work done by a magnetic field on a moving charge is zero since the magnetic force is perpendicular to the velocity.
2. However, magnetic forces in a multiparticle system can rearrange the energy inside the system if these forces do as much negative work as they do positive work, resulting in a net work done of zero.

Mathematical expression:

1. The work done is $W=\int \overrightarrow{F.}\overrightarrow{dl}$, where $\overrightarrow{F}$ is the force done on an object and $\overrightarrow{dl}$ is the displacement.
2. In the case of a Magnetic field with $B$ and velocity $v$, with a charged particle $q$the force is $\overrightarrow{F}=q\overrightarrow{V}\times \overrightarrow{B}$.
3. If we calculate the work done by magnetic force

$$\begin{gathered} W=\int \overrightarrow{F.}\overrightarrow{dl} \\ \Rightarrow \int (q\overrightarrow{V}\times \overrightarrow{B}).\overrightarrow{dl} \\ \Rightarrow \int (q\overrightarrow{V}\times \overrightarrow{B}).\overrightarrow{V}dt \\ \Rightarrow \int qdt[\overrightarrow{V}\times \overrightarrow{B}].\overrightarrow{V} \end{gathered}$$ and we know that $\overrightarrow{V}=\frac{\overrightarrow{dl}}{dt}\Rightarrow \overrightarrow{dl}=\overrightarrow{V}dt$

The cross-product of two identical vectors is zero.

Thus, we can observe from the above equation that the work done from the magnetic force is zero.