Figure shows a straight, long wire carrying a current i and a rod of length ℓ coplanar with the wire and perpendicular to it. The rod moves with a constant velocity v in a direction parallel to the wire. The distance of the wire from the centre of the rod is x. Find the motional emf induced in the rod.

Given:

Current = i

Length of rod = l

Velocity = v

Distance of centre of rod from wire = x

Hence, the two ends of the rod are at distances and from the wire

Diagram:

Formula used:

We consider an element of length da at a distance of ‘a’ from the wire.

Now, magnetic field due to an infinite current carrying wire at a distance a … (i), where μ₀ = magnetic permeability of vacuum, i = current, a = distance from wire.

Therefore, emf induced in the element … (ii), where B = magnetic field, da = element, v = velocity

Hence, putting (i) in (ii), we get

Integrating on both sides and putting suitable limits, we get

⇒ (ans)