Question

A wire, carrying a current i, is kept in the x−y plane along the curve y = A sin `((2x)/lamda x)`. 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 ((2x)/lamda x)` 
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
`vecF = i veclxxvecB`
For a small element dl,
`dvecF = i (dl xx vecB)`
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 = iB \[\int\limits_0^{λ}\] dl
`⇒ F = ilamdaB`