Derive an expression for magnetic force $\displaystyle \overrightarrow{\mathrm{F}}$ acting on a straight conductor of length L carrying current I in an external magnetic field $\displaystyle \overrightarrow{\mathrm{B}}$. Is it valid when the conductor is in zig-zag form ? Justify.

Total number of mobile charge carriers in a conductor of length L, cross-sectional area A and number density of charge carriers n: $\displaystyle =n L A$ Force acting on the charge carriers in external magnetic field $\displaystyle \vec{B}$: $\displaystyle \vec{F}=(n A L) q \vec{v}_{d} \times \vec{B}$ − − − − − −($\displaystyle 1$) where $\displaystyle v_{d}$ is the drift velocity of the charge carriers. Current flowing: $\displaystyle I=v_{d} q n A$ $\displaystyle I L=v_{d} q n A L$ − − − − − −($\displaystyle 2$) On solving equations ($\displaystyle 1$) and ($\displaystyle 2$): $\displaystyle \vec{F}=I(\vec{L} \times \vec{B})$ Yes, because this force can be calculated by considering the zig-zag conductor as a collection of linear strips $\displaystyle (d \vec{l})$ and summing them vectorially.