A wire of length l in the form of a square loop is in a plane normal to magnetic field B. If the length of the wire is decreasing at the rate dl dt , the e.m.f. induced is e₁ . If this wire is converted into a circular loop, the e.m.f. induced in a circular loop is e₂ . Then e₁ e₂ is

Step-by-step solution

For the square loop, the side length is a = l 4 . Area of the square loop is A₁ = a^2 = l^2 16 . The induced e.m.f. e₁ is given by e₁ = | d ₁ dt | = B dA₁ dt = B d dt ( l^2 16 ) = Bl 8 | dl dt | . For the circular loop, the circumference is l , so the radius is r = l 2 . Area of the circular loop is A₂ = r^2 = ( l 2 )^2 = l^2 4 . The induced e.m.f. e₂ is given by e₂ = | d ₂ dt | = B dA₂ dt = B d dt ( l^2 4 ) = Bl 2 | dl dt | . Taking the ratio of the two induced e.m.f.s: e₁ e₂ = Bl 8 | dl dt | Bl 2 | dl dt | = 2 8