Two long thin concentric perfectly conducting cylindrical shells of radii $a$ and $b$ $(a<b)$ are connected together at one end by a resistor of resistance $R$, and at the other by a battery that establishes a potential difference $V$. Thus, a current $I=V / R$ flows in opposite directions along each of the cylinders.

(a) Using Ampère's law, find the magnetic field $\mathbf{B}$ in between the cylinders.

(b) Using Gauss's law and the integral relationship between the potential and the electric field, or otherwise, show that the charge per unit length on the inner cylinder is

$$\lambda=\frac{2 \pi \epsilon_{0} V}{\ln (b / a)}$$

and also calculate the radial electric field.

(c) Calculate the Poynting vector and by suitable integration verify that the power delivered by the system is $V^{2} / R$.