Question

Two equal and opposite point charges, q and -q are located at: (0, 0, d/2) and (0, 0, -d/2), respectively as shown in the figure below. This arrangement is known as an electric dipole. Use first-order Taylor expansions to show that the total electric field due to this electric dipole at very large distances from the origin as compared to the dipole separation, d, is given approximately by: qd t+q d/2 | θ -d/2 -q

Answer 1

due to for away point p above all theta and R are due to these + q and -q potential can be written V = V_+ + V_- V = q/4 pi epsilon_0 R + + -q/4 pi epsilon_0 R - Note due to point charge at distance R potential defined V = Q/4 pi epsilon_0 R since point p is far away therefore R_+ = R - d/2 cos theta R_- = R + d/2 cos theta Hence potential equation become V = q/ 4 pi epsilon_0 [1/R - d/2 cos theta - 1/R + d/2 cos theta V = q/4 pi epsilon_0 [R + d/2 cos theta - (R - d/2 cos theta)/R^2 - d^2/4 cos^2 theta Since R > > d therefore R^2 > > d^2/4 cos^2 theta \r\n Hence V = qd cos theta/4 pi epsilon_0 R^2: [R > > d] Now Electric field is defined as E = - = -(partial deferential V/ partial deferential R qR + 1/R partial deferential V/ partial deferential theta a theta E = -qd/ 4 pi epsilon_0 [cos theta partial deferential/partial deferential R