(Solved): 2. Consider two parallel metal discs of radius a with a small distance d apart, with charges ...

2. Consider two parallel metal discs of radius $$a$$ with a small distance $$d$$ apart, with charges $$\pm Q$$ in the lower/upper disc. Let the axis of the discs be the $$z$$ coordinate, so that the initial electric field between the two discs is $$\mathbf{E}=\frac{Q}{{?}_{0}A}\hat{z}$$. Now, the two discs are connected by a resistive wire placed at the central axis between the discs, so that the current $$I(t)=?\frac{dQ(t)}{dt}$$ in the $$z$$ direction discharges the discs. (a) Compute the initial energy stored in the electric field. (b) From the time-varying electric field, $$\mathbf{E}(t)=\frac{Q(t)}{{?}_{0}A}\hat{\mathbf{z}}$$, compute the displacement current density $$J_d={?}_0\frac{dE}{dt}$$ in terms of $$I(t)$$. Together with the physical current $$I(t)$$ in the wire at the center, it produces a magnetic field $$\mathbf{B}(s,t)$$ in the region between the discs, where $$s$$ is the distance from the axis. Find $$\mathbf{B}(s,t)$$, and show that $$\mathbf{B}(s,t)=0$$ for $$s>a$$. (c) Compute the Poynting vector $$\mathbf{S}(s,t)$$. Compute the total power inflow to the wire $$P(t)=?\underset{s?0}{\lim }{?}_{?(s)}\mathbf{S}(t)?d\mathbf{a}$$, where $$?(s)$$ is the small cylinder of radius $$s$$ and length $$d$$ enclosing the wire. Integrating $$P(t)$$ in time, show that the total energy absorbed by the wire is equal to the initial energy stored in the electric field that is found in (a).