A Dipole in an Electric Field — Question 1

Question 1

A uniform field exerts a couple on a neutral rigid dipole

A rigid dipole consists of charges $-q$ and $+q$ separated by vector $\mathbf d$ directed from negative to positive, so $\mathbf p=q\mathbf d$. The external field is uniform. All torques are about the midpoint.

1. Derive the net force and vector torque from the two individual forces.

2. For $\mathbf p=(2.00,3.00,0)\times10^{-9}$ C m and $\mathbf E=(400,0,0)$ N/C, find the torque and its rotational sense as viewed from positive z.

Question 1 - Solution

A uniform field exerts a couple on a neutral rigid dipole

(a) Method and calculation

$$\mathbf F_+=q\mathbf E,\quad\mathbf F_-=-q\mathbf E,\quad\mathbf F_{\mathrm{net}}=\mathbf0.$$$$\boldsymbol\tau=\frac{\mathbf d}{2}\times q\mathbf E+\left(-\frac{\mathbf d}{2}\right)\times(-q\mathbf E)=\mathbf p\times\mathbf E.$$

(b) Interpretation and further calculation

$$\tau_z=p_xE_y-p_yE_x=-(3.00\times10^{-9})(400).$$$$\boldsymbol\tau=-1.20\times10^{-6}\hat{\mathbf z}\,\mathrm{N\,m}.$$The negative-z torque is clockwise when viewed from positive z toward the origin. It tends to align the moment with the field.

Physical check

Zero net force permits a nonzero torque because the equal and opposite forces act on different lines.