[2018EM/HMW-04]

Electrodynamics                                                                         May 8, 2022

Tutorial IV

1. A capacitor is made up of two plane parallel plates of width \(a\) and length \(b\) separated by a distance \(d\). The capacitor has a dielectric slab of relative dielectric constant \(K\) between the two plates.
   1. The capacitor is connected to a battery of emf \(V\). The dielectric slab is partially pulled out of the plates such that only a length \(x\) remains between the plates. Calculate the force on the dielectric slab which tends to pull it back into the place.
   2. With the dielectric plate fully inside, the capacitor plates are charged to a potential \(V\) and the battery is disconnected. Again, the dielectric slab is pulled out such that only a length \(x\) remains inside the plates. Calculate the force on the dielectric slab which tends to pull it back into the plates.
   Neglect the effect of edges in both parts (a) and (b).
2. Consider a point charge \(q\) embedded in a semi-infinite medium of dielectric constant \(\epsilon_1\), and located at a distance \(d\) from a plane interface that separates the first medium from another semi-infinite dielectric medium of dielectric constant \(\epsilon_2\). Assume that the interface coincides with the plane \(z=0\). Obtain the electric potential in the two regions. You must not use the image charge values and locations discussed in the class. So part of the problem for you is to investigate if the boundary conditions on the interface determine the location and value of the image charge uniquely and completely.
3. Consider a homogeneous dielectric \(\epsilon\) of infinite extent, in which there is a uniform field \(E_0\). A spherical cavity of radius \(a\) is cut out of the dielectric. Find
   1. Potential in the cavity and its surface
   2. The polarization charge density on the walls.
   3. The field outside the cavity
4. A point charge \(q\) and a dielectric plate of permittivity \(\epsilon_2\) are both in a dielectric medium of permittivity \(\epsilon_1\). Find sequence of images and their images and so on which will solve the problem. Obtain the solution for the potential in the three regions as an infinite series.
5. A point charge is placed at a distance \(d\) from a grounded conducting infinite plane.
   1. Find an expression for surface charge density induced on the plane. Use your expression for surface charge density to find the force of attraction on the point charge.
   2. Find the electrostatic energy of the system by computing the work done in bringing the point charge from infinity to its position at a distance \(d\).
   3. Obtain the electrostatic energy of the system by computing the integral of energy density \(\frac{1}{2} \epsilon_0 |\vec{E}|^2\). Use this answer to obtain the force of attraction on the point charge.