Question

2. Electric dipole (20%) A very common charge arrangement in electromagnetics is the dipole. A charge dipole is illustrated in the figure below. In this problem, we will find the potential and field due to a dipole. tq X (a) Find the electrostatic potential at an arbitrary position in spherical coordinates, that is find the potential at the position (r, 0,0). (b) For large values of r, we can write: Z(17 cos 0 + -1/2 ~(1 7. cos 6)-1/2 ~ (1+ cose). Explain the origins of these approximations and why they are valid for large r. (c) Use the approximations described in part (b) to formulate a simplified version of your expression in part (a) for large r. How does this expression differ from that which would follow from a single charge (+9) placed at the origin? (d) Using your expression from part (c), calculate the electric field due to the dipole at large values of r. (e) Use your answer from part (d) and your intuition to sketch the electric field lines due to the dipole.

Answer 1

*9 the electre potential is given by: V=VatVe=K9, +42 a = 1145247294)rous go) e " -q* (Eur'de (0)" (eur's dreces ] @ A=(4, 75, diraso5l Verdre, tregon veutta sead for rond son o-4474.colo) 444 44.00) = 1 con co - i => VE KAd cojo © Aamerikan Phot c = -2V=-1R_Vi troverete B) bu - lk Podcaso of exponsion. Oo aus - Kqd sine > ē = 2uqd coso es + Kg dine ê, r3 r2