The Electric Field Due to an Electric Dipole — Question 3

Question 3

An axial approximation needs a quantified distance criterion

For the finite dipole at $x>a$, use $E_{\mathrm{exact}}=(2kp/x^3)(1-a^2/x^2)^{-2}$ and approximation $E_d=2kp/x^3$. Define relative error as $(E_{\mathrm{exact}}-E_d)/E_{\mathrm{exact}}$.

1. Derive the error as a function of $u=a/x$.

2. Find the minimum $x/a$ for error no greater than 1.00 percent and identify whether the approximation overestimates or underestimates the exact field.

Question 3 - Solution

An axial approximation needs a quantified distance criterion

(a) Method and calculation

$$\epsilon=1-\frac{E_d}{E_{\mathrm{exact}}}=1-(1-u^2)^2=2u^2-u^4.$$$$\epsilon\le.0100\Rightarrow(1-u^2)^2\ge.9900.$$

(b) Interpretation and further calculation

$$0<u<1:\quad u^2\le1-\sqrt{.9900}.$$$$x/a\ge[1-\sqrt{.9900}]^{-1/2}\approx14.1244.$$The exact multiplier exceeds one, so the point-dipole approximation underestimates the axial field.

Physical check

An error percentage is ambiguous unless its denominator is stated.