Equations E= 1/2 pi epsilon_0 qd/z^2 and E = 1/2 pi epsilon _0 P/z^3 are approxi
ID: 1527742 • Letter: E
Question
Equations E= 1/2 pi epsilon_0 qd/z^2 and E = 1/2 pi epsilon _0 P/z^3 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 = 4.50d from the dipole center (where d is the separation distance between the particles of the dipole). Let E_appr be the magnitude of the field at point P as approximated by E = 1/2 pi epsilon_0 qd/z^3 and E =1/2 pi epsilon_0 P/z^3 (electric dipole). Let E be the actual magnitude. By how much is the ratio E_appr/E_act less than 1?Explanation / Answer
Actual electric field on the axis,
E = E(+q) - E(-q)
= k*q/(4.5*d - d/2)^2 - k*q/(4.5*d + d/2)^2
= k*q/(16*d^2) - k*q^2/(25*d^2)
= 0.0225*k*q/d^2
Eappr = (1/(2*pi*epsilon))*(q*d)/z^3
= 2*k*q*d/z^3
= 2*k*q*d/(4.5*d)^3
= 0.02195*k*q/d^2
so,
Eappr/Eact = 0.02195/0.0225
= 0.9756
= 1 - 0.0244
so, the ratio is 0.0244 less than Eappr/Eact
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