Question

A wire, carrying a current i, is kept in the x−y plane along the curve $y=A\sin \left(\frac{2\mathrm{\pi }}{\lambda }x\right).$ A magnetic field B exists in the z direction. Find the magnitude of the magnetic force on the portion of the wire between x = 0 and x = λ.

Answer 1

Given:
Electric current flowing through the wire = i
The wire is kept in the x−y plane along the curve, $y=A\sin \left(\frac{2\mathrm{\pi }}{\lambda }x\right)$
Magnetic field (B) exists in the z direction.

We have to find the magnetic force on the portion of the wire between x = 0 and x = λ.
Magnetic force is given by
$\overrightarrow{F}=i\overrightarrow{l}\times \overrightarrow{B}$
For a small element dl,
$d\overrightarrow{F}\mathit{=}i\left(\mathrm{d}l\mathit{\times }\overrightarrow{B}\right)$

The effective force on the whole wire is equivalent to the force on a starlight wire of length λ placed along the x axis.
So,
$F\mathit{=}iB{\int }_0^{\lambda }dl$
$\Rightarrow F=i\lambda B$