[UPSC CSE 2008] Electrostatics and Magnetostatics (Q187, 10M)

State Faraday's Law of induction in terms of emf and magnetic flux. Use Stoke's theorem to express the law in differential form. A uniform, time varying magnetic field $ \bar{B} = \bar{B}_0(1 + \alpha t)\hat{k} $ fills a circular region of radius R lying in the x-y plane with the centre at origin. Here $ \bar{B}_0 $ and $ \alpha $ are appropriately dimensioned constants. Find the induced electric field at a distance r from the centre where r < R.