Question

A conducting sphere with radius R is centered at the origin. The sphere is grounded having an electric potential of zero. A point charge Q is brought toward the sphere along the z- axis and is placed at the point ะ-8. As the point charge approaches the sphere mobile charge is drawn from the ground into the sphere. This induced charge arranges itself over the surface of the sphere, not in a uniform way, but rather in such a way that the potential throughout the sphere remains zero. In the space outside the sphere the point charge and the induced charge both contribute to the net electric potential which will vary from point to point. As a function of the radial distance from the origin and the angle φ from the z-axis, the potential is given in both regions by the following 2 r<R rR Verify that this function is continuous at r- R. The radial d ular components of the electric field are given by OV Compute these components and verify that the electric field i conductor by evaluating Er and Eon the surface, r-R. perpendicular to the surface of the The surfacc charge density on a conductor is given by the following where n is the unit vector perpendicular to the conductor's surface pointing away from the conductor (the normal). In the case described above, the normal is f and to evaluate the expression on the surface is to evaluate it with r- R. The following shows these steps Using the process with the electric field determined above, compute σ(d) The total charge on a surface can be evaluated by integrating the surface charge density over the area, In this case, the total charge on the surface is the charge drawn to the sphere from the ground when the point charge is brought near the sphere. This is called the induced charge Using the surface charge found above and the differential area given by, compute the induced charge on the sphere

Answer 1

Given, V(r, emptyset) = 0 r < R So Lt_r rightarrow R^- (r, emptyset) = 0 similarly for r > R v(r, emptyset) = Q/4 pi epsilon_0 (s^2 + r^2 - 2 s r cos beta)^1/2 - Q/4 pi epsilon_0 (s^2 r^2/R^2 + R^2 - 2 s r cos emptyset)^1/2 pully r = R in above v_r rightarrow R^+ (r, emptyset) = 0 So it is continuous as v_r rightarrow R^+ (r, emptyset) = v_r rightarrow R^- (r, emptyset) E_r = -partial differential v/partial differential r = -1/2 Q(2r - 2 s cos emptyset)/4 pi epsilon_0 (s^2 + r^2 - 2 s r cos emptyset)^3/2 - -1/2 Q(2 r s^2/R^2 - 2 s cos emptyset)/4 pi epsilon_0 (s^2 r^2/R^2 + R^2 - 2 s r cos emptyset)^2 E_r at r = R E_r = -1/2 Q(2R - 2 s cos emptyset)/4 pi epsilon_0 (s^2 + R^2 - 2 S R cos emptyset)^3/2 - -1/2 Q(s^2/R - 2 s cos emptyset)/4 pi epsilon_0 (s^2 + R^2 - 2 s R cos emptyset)