Equations 22-8 and 22-9 are approximations of the magnitude of the electric fiel

Question

Equations 22-8 and 22-9 are approximations of the magnitude of the electric field of an electric dipole, at points along the dipole axis. Consider a point P on that axis at distance z = 9.00 d from the dipole center(where d is the separation distance between the particles of the dipole). Let Eappr be the magnitude of the field at point P as approximated by Equations 22-8 and 22-9. Let E act be the actual magnitude. By how much is the ratio Eappr/Eact less than 1? 0.99 E = q/4 pi epsilon 0 z2 2d/z = 1/2 pi epsilon 0 qd/z3 (22-8) E = 1/2 pi epsilon 0 P/z3 (electric dipole). (22-9)

Explanation / Answer

Eact = (1/40)*[q/(z -d/2)2 - q/(z + d/2)2] =(1/40)*2qdz/[z2 -(d/2)2]2 =(P/20)*z/[z2 -(d/2)2]2 Eapp = (P/20)/z3 Eapp/Eact = [z2 -(d/2)2]2/z4 = [1- (d/2z)2]2 = [1 -(1/18)2]2 the required value = 1 - [1 - (1/18)2]2 =0.0062

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