9. Supplementary Questions

9.1 A sphere of radius \(2R\) is made of a non-conducting material that has a uniform volume  charge density \(\rho\). A spherical cavity of radius R is then carved out at a point R from the centre of the sphere. Compute the electric field within the cavity.

Idea: Kickstart your thinking to similar questions, where more variations could occur. The cavity could be uniformly filled with charge density -\(\rho\)or filled with a dielectric, and so on. 

9.2 Consider an insulating chessboard of d x d, where every black tile has \(\sigma\) surface charge spread evenly on the surface, and every white tile has no charge on it. Determine the electric field at a distance d/2 above the centre of the chessboard.

Idea: This is perhaps my favourite question in electrostatics. Simple in construction, yet unable to be brute forced, and the elegant solution requires a combination of many simple concepts. Hint: Consider symmetry and Gauss' Law.

9.3 An uncharged spherical conductor has a odd-shaped cavity, within which consists a charge q. Determine the field outside the sphere.

Idea: Although simple application of Gauss' Law gives you the required result, it is the fact that neither the position of the charge nor the shape of the cavity affects the final result that I think makes this question worth examining.

9.4 In a vacuum diode, electrons are "boiled" off a hot cathode, at potential zero, and accelerated across a gap to the anode, which is held at positive potential \(V_0\). The cloud of electrons within the gap (called space charge) quickly builds up to the point where it reduces the field at the surface of the cathode to zero. From then on a steady current \(I\) flows between the plates. Suppose the plates are large relative to their separation. Then, V,  \(\rho\) (volume charge density), and v (the speed of electrons) are all functions of x alone.

9.4.1 Assuming the electrons start from rest at the cathode, what is their speed at point x, where the potential is V(x)?

9.4.2 In the steady state, I is independent of x. What, then, is the relationship between  \(\rho\) and v?

9.4.3 Use these results, and assuming that \(\frac{d^2V}{dx^2}=\frac{\rho}{\epsilon}\), to obtain a differential equation for V, by eliminating \(\rho\) and v.

9.4.4 Solve this equation for V as a function of x, \(V_0\), and d. Plot V(x), and compare it to the potential without space-charge. Also, find  \(\rho\) and v as functions of x.

9.4.5 Show that