A conducting rod of length L lies in X Y -plane and makes an angle 30^ with X -axis. One end of the rod lies at origin initially. A magnetic field also exists in the region pointing along positive Z -direction. The magnitude of the magnetic field varies with y as B₀ ( y L )^3 , where, B₀ is a constant. At some instant the rod starts moving with a velocity v₀ along X -axis. The emf induced in the rod is

Step-by-step solution

Length along Y -axis would be l 30^ = l 2 length =L Velocity of rod =v₀ , along X -axis variable magnet is field, B=B₀ ( y L )^3 , along Z -axis. Now, induced emf (formula) E=B l v where, B, l_ and v are perpendicular to each other, l=L / 2 Now, consider a small element of length d y at a distance y from origin along Y -axis, then emf of element d E=B(d y) v₀=B₀ y^3 L^3 v₀ d y Emf of the rod, aligned & E= B₀ v₀ L^3 ₀^ L / 2 y^3 d y & E= B₀ v₀ L^3 y^4 4 ^ L / 2 & E= B₀ v₀ L^3 (L / 2)^4 4 E= B₀ v₀ L 64 aligned