Question

(3) An electric dipole consists of two point charges with opposite signs separated by a distance 2a. Assume the dipole is along the x-axis. Show that the electric field at a distant point on the dipole axis can be approximated by~ 4keqa/x

Answer 1

E_+ = k Q/x^2 (To the right) E_- = k Q/(x - 2a)^2 (to the left) E_net = E_- - E_+ = k Q [1/(x - 2a)^2 - 1/x^2] E_net = k Q [1/x^2 (1 - 2a/x)^2 - 1/x^2] E_net = k Q/x^2 [1/(1 - 2a/x)^2 - 1] E_net = k Q/x^2 [(1 - 2a/x)^-2 - 1] E_net almostequalto k Q/x^2 [1 + 4a/x - 1] (since 2a/x < < 1) E_net almostequalto 4 k Q a/x^3

$E_&plus;=\frac {kQ}{x^2}$ (To the right)

$E_-=\frac {kQ}{(x-2a)^2}$ (To the left)

$E_{net}=E_--E_&plus;=kQ\left [\frac {1}{(x-2a)^2}-\frac {1}{x^2} \right ]$

$E_{net}=kQ\left [\frac {1}{x^2(1-\frac {2a}{x})^2}-\frac {1}{x^2} \right ]$

$E_{net}=\frac {kQ}{x^2}\left [\frac {1}{(1-\frac {2a}{x})^2}-1 \right ]$

$E_{net}=\frac {kQ}{x^2}\left [(1-\frac {2a}{x})^{-2}-1 \right ]$

$E_{net}\approx\frac {kQ}{x^2}\left [1&plus;\frac {4a}{x}-1 \right ]$ (Since $\frac {2a}{x}<<1$ )

$E_{net}\approx\frac {4kQa}{x^3}$