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# 3.5: Force on a Dipole in an Inhomogeneous Electric Field

$$\text{FIGURE III.4}$$

Consider a simple dipole consisting of two charges $$+Q$$ and $$-Q$$ separated by a distance $$δx$$, so that its dipole moment is $$p = Q\, δx$$. Imagine that it is situated in an inhomogeneous electrical field as shown in Figure $$III$$.4. We have already noted that a dipole in a homogeneous field experiences no net force, but we can see that it does experience a net force in an inhomogeneous field. Let the field at $$−Q \text{ be }E$$ and the field at $$+Q \text{ be }E + δE$$. The force on $$−Q \text{ is }QE$$ to the left, and the force on $$+Q \text{ is }Q(E + δE)$$ to the right. Thus there is a net force to the right of $$Q\, δE$$, or:

$\label{3.5.1}\text{Force}=p\frac{dE}{dx}$

Equation \ref{3.5.1} describes the situation where the dipole, the electric field and the gradient are all parallel to the x-axis. In a more general situation, all three of these are in different directions. Recall that electric field is minus potential gradient. Potential is a scalar function, whereas electric field is a vector function with three component, of which the x-component, for example is $$E_x=-\frac{∂V}{∂x}$$. Field gradient is a symmetric tensor having nine components (of which, however, only six are distinct), such as $$\frac{∂^2V}{ ∂x^2},\,\frac{ ∂^2V}{ ∂y ∂z}$$ etc. Thus in general equation \ref{3.5.1} would have to be written as

$\begin{pmatrix}E_x \\ E_y \\ E_z \\ \end{pmatrix} =-\begin{pmatrix}V_{xx} & V_{xy} & V_{xz} \\ V_{xy} & V_{yy} & V_{yz} \\ V_{xz} & V_{yz} & V_{zz} \\ \end{pmatrix}\begin{pmatrix}p_x \\ p_y \\ p_z \\ \end{pmatrix}\label{3.5.2}$

in which the double subscripts in the potential gradient tensor denote the second partial derivatives.