3. A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap in the wire, of width w a, forms a parallel-plate capacitor as shown in Fig. 2. Assume the charge is zero at t = 0. Calculate the Poynting vector in the gap, as functions of the distance s from the axis and the time t.

4. As shown in Fig. 3, a very long solenoid of radius \( a \), with \( n \) turns per unit length, carries a current \( I_s \). Coaxial with the solenoid, at radius \( b >> a \), is a circular ring of wire, with resistance \( R \). When the current in the solenoid is gradually decreased, a current \( I_c \) is induced in the ring. The Poynting vector just outside the solenoid is \[ \vec{S} = \frac{\vec{E} \times \vec{B}}{\mu_0} = - \frac{1}{4} \mu_0 I_s \frac{dI_s}{dt} \frac{a^2 n}{(b^2 + z^2)^{3/2}} \hat{r}. \] Calculate the power delivered to the ring \( = R I_c^2 \). (Hint: You need to calculate an integral and use \( \int \frac{dz}{(b^2 + z^2)^{3/2}} = \frac{z}{b^2(b^2 + z^2)^{1/2}} \))

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