Figure shows a conducting circular loop of radius a placed in a uniform, perpendicular magnetic field B. A thick metal rod OA is pivoted at the centre O. The other end of the rod touches the loop at A. Tec entre O and a fixed point C on the loop are connected by a wire OC of resistance R. A force is applied at the middle point of the rod OA perpendicularly, so that the rod rotates clockwise at a uniform angular velocity ω. Find the force.

Question

Figure shows a conducting circular loop of radius a placed in a uniform, perpendicular magnetic field B. A thick metal rod OA is pivoted at the centre O. The other end of the rod touches the loop at A. Tec entre O and a fixed point C on the loop are connected by a wire OC of resistance R. A force is applied at the middle point of the rod OA perpendicularly, so that the rod rotates clockwise at a uniform angular velocity ω. Find the force.

Philoid · Accepted Answer

Given:

Radius = a

Magnetic field = B

Resistance = R

Angular velocity = w

Formula used:

Let us consider an element of length dr at a distance r from the centre.

Hence, induced emf on this portion … (i), where B = magnetic field, dr = length of element, w = angular velocity, r = distance from centre (since v = ωr)

Hence, integrating on both sides with suitable limits, we get

⇒

Now, current , where E = emf, resistance = R

=>

Hence, force on the rod (where I = current, a = length of rod, B = magnetic field) ⇒ (Ans)