The Electric Field Due to an Electric Dipole — Question 9

Question 9

A neutral system has an origin-independent dipole moment

For discrete charges define $\mathbf p=\sum_iq_i\mathbf r_i$. Shift the coordinate origin by vector $\mathbf b$, so $\mathbf r_i\prime=\mathbf r_i-\mathbf b$.

1. Derive the transformation of the dipole moment and state the condition for invariance.

2. Apply it to $-q$ at $x=0$ and $+q$ at $x=d$, then to $-q$ at $x=0$ and $+2q$ at $x=d$, when the origin is shifted to $x=d/2$.

Question 9 - Solution

A neutral system has an origin-independent dipole moment

(a) Method and calculation

$$\mathbf p\prime=\sum_iq_i(\mathbf r_i-\mathbf b)=\mathbf p-\mathbf b\sum_iq_i.$$$$\mathbf p\prime=\mathbf p-Q_{\mathrm{total}}\mathbf b,\quad Q_{\mathrm{total}}=0\Rightarrow\mathbf p\prime=\mathbf p.$$

(b) Interpretation and further calculation

$$(-q,+q):\quad p_x=qd,\quad p_x\prime=qd.$$$$(-q,+2q):\quad p_x=2qd,\quad p_x\prime=2qd-q(d/2)=\tfrac32qd.$$A charged system still has a definable moment about a chosen origin, but it lacks the origin independence of a neutral dipole.

Physical check

The physical field does not depend on the coordinate origin; a complete expansion changes its other terms consistently.