Inductive battery chargers, which allow transfer of electrical power without the need for exposed electrical contacts, are commonly used in appliances that need to be safely immersed in water, such as electric toothbrushes. Consider the following simple model for the power transfer in an inductive charger (see the figure). Within the charger's plastic base, a primary coil of diameter d with n_{\rm{P}} turns per unit length is connected to a home's ac wall outlet so that a current I = I_0sin(2\pift) flows within it. When the toothbrush is seated on the base, an N-turn secondary coil inside the toothbrush has a diameter only slightly greater than d and is centered on the primary.
Find an expression for the emf induced in the secondary coil. [This induced emf recharges the battery.]