A Dipole in an Electric Field — Question 7

Question 7

An inclined dipole in a saddle field experiences both translation and rotation

An ideal permanent point dipole is centered at $(x,y)=(0,b)$ in the static field $\mathbf E=(Gx,-Gy,0)$, with $G,b>0$. Its instantaneous moment is $\mathbf p=p(\hat{\mathbf x}+\hat{\mathbf y})/\sqrt2$. Use $U=-\mathbf p\cdot\mathbf E$ and hold orientation fixed when taking spatial derivatives.

1. Derive the general force for fixed components $(p_x,p_y)$ and evaluate it for the stated orientation.

2. Calculate the torque about the center and explain why zero total charge does not imply zero force.

Question 7 - Solution

An inclined dipole in a saddle field experiences both translation and rotation

(a) Method and calculation

$$U=-G(p_xx-p_yy),\quad\mathbf F=-\boldsymbol\nabla U=(Gp_x,-Gp_y,0).$$$$\mathbf F=\frac{Gp}{\sqrt2}(1,-1,0).$$

(b) Interpretation and further calculation

$$\mathbf E(0,b)=(0,-Gb,0),\quad\tau_z=p_xE_y-p_yE_x.$$$$\boldsymbol\tau=-\frac{pGb}{\sqrt2}\hat{\mathbf z}.$$The field differs between the two constituent charge locations, leaving a net force in the dipole limit. The local field also tends to turn the moment clockwise.

Physical check

The partial derivatives act on the external field, not on a moment allowed to reorient during the derivative.