Consider the configuration shown in the figure: An in- finite plane conductor has a small hemispherical knob of radius a as indicated. A point charge q is placed directly to the right hand side of the hemispherical knob at a distance b from its center. Assume that the boundary, i.e. the infinite plane with the knob, is kept at zero potential. a) Find the potential p at a generic point P on the right hand side. b) Determine the appropriate Dirichlet Green function for this geometry and write down the most general solution of the potential p given a generic charge density p on the right hand side of the boundary and that potential p = f("r) on the boundary. %3D c) Find the total charge induced on the knob. P(r) a b

Please answer f only

Consider a conductor plate carrying a current I as shown in the figure. Current flows in the negative direction. It is also known that a conductor has a charge-carrying density of n with 1 particle carrying a charge q . This conductor plate is then placed on the a region with an external magnetic B field in the z direction . Determine the potential difference across the conductors! State the answer in terms of B, n, q, I, B, and d! (Assume positive charge carrier) *This potential, which is often referred to as the Hall potential V_H, is commonly measured in experiments to determine the properties of a material. However, there are other quantities that are also measured namely the Hall coefficient R_H. In ordinary metals, the relationship between R_H and V_H is as following (equation in figure)

The rectangular loop shown in Figure (Q2) is coplanar with a long, straight wire carrying a current (I). The loop has two sides of length (L) parallel to the wire, one at distance (a) and the other at a distance (b) from the wire. Find the magnetic flux through the loop. I a

Q2. A toroidal core that is made of iron and having relative permeability of µ, = 5000. The mean path length of the core, le is 37.6 cm while the cross-sectional area of the core is 0.1256 cm². The fringing effect and leakage flux are to be negligible. The number of turns of the coil, N¡ is 400. If there is an airgap of 5 mm and magnetic flux density at air gap is 0.25 T, (i) Draw equivalent magnetic circuit analogy of the core. (ii) Calculate the current required to energize the core. (iii) Determine the magnetizing strength, Hg in the air gap. mark (iv) If flux density in the air gap is reduced to half, find the new current flowing in the coil to energize the core. (v) If there is no airgap and the current value is similar as in (ii), what is the flux density in the core ? (Note: lo = 4n x10-7 H/m, Bmax = 1.5 T).