Question

FIG. 2. Setup of Exercise 3 Exercise 3 The electrostatic potential of an electic dipole moment d located at the origin takes the following form d-T Tr where r is the vector joining the origin to the point X (7 is called the "position vector" in the textbook). See Fig. 2 (i) Chosing the z axis to be aligned with the electric dipole moment, express φ in terms of cartesian, cylindrical, (ii) The electric field is obtained from E-- Compute E in all three coordinate systems (iii) Compute the divergence of E in all three coordinate systems. Make sure you get the same answer (iv) Assuming d has a unit magnitude, draw the vector E at the following 20 points in the y - z plane (i.e. all of and spherical polar coordinates them have π/2), whoose coordinates (r, θ) are given by (7,9)=(1.44, nn/6), n=0,1,2, ,11

Answer 1

An electric dipole moment consists of two equal and opposite charges +q and -q separated by a distance The electric potential at a point 'p� with position vector r^rightarrow is given by phi (r^rightarrow) = 1/4 pi epsilon_0 [q/r_1 - q/r_2] = q/4 pi epsilon_0 [r_2 - r_1/r_1 r_2] -(1) in polar the dipole moment p^rightarrow is given by d^rightarrow = d cos theta r^^- d sin theta theta^^-(2) where r^^and theta^^are unit vectors in radial and tangential direction Now if a similarto small compared to 'r�, we may use r_2 - r_1 almostequalto a cos theta r_1 r_2 almostequalto r^2 where theta is angle b/w r^rightarrow and dipole moment vector d^rightarrow This gives phi (r^^) almostequalto q a cos theta/4 pi epsilon_0 r^2 = d cos theta/4 pi epsilon_0 r^2 Cartesian Coordinates Let the dipole moment vector be taken along z- axis and position vector r^rightarrow of d in y - z plane then WR have cos theta