Question

Assuming an equation of magnetic force due to arbitrarily shaped wire, obtain an expression for force on a closed circuit in a uniform magnetic field.

Answer 1

The magnetic force on the charge dq is

`vecf_m = dq (vecv_d xx vecB )` 

= `dq (vec(dl)/vec(dt) xx vecB)` 

= `I (vec(dl) xx vecB )    ...(∵ I = (dq)/(dt))`

The maximum value of the force is F_max = IlB, when sin θ = 1, that is, when the conductor lies at right angles to `vecB` (θ = 90°).

For a current-carrying conducting loop (closed circuit) in a uniform field, 

`vecF = intvecf_m = I (oint vec(dl) × vecB)` 

But for a closed loop of arbitrary shape, the integral is zero. 

∴ `vecF = 0`