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Physics LibreTexts

3.5: Force on a Dipole in an Inhomogeneous Electric Field

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III.4.pngFIGURE 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=Qδx. 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 be E and the field at +Q be E+δE. The force on Q is QE to the left, and the force on +Q is Q(E+δE) to the right. Thus there is a net force to the right of QδE, or:

Force=pdEdx

Equation ??? 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 Ex=Vx. Field gradient is a symmetric tensor having nine components (of which, however, only six are distinct), such as 2Vx2,2Vyz etc. Thus in general Equation ??? would have to be written as

(ExEyEz)=(VxxVxyVxzVxyVyyVyzVxzVyzVzz)(pxpypz)

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


This page titled 3.5: Force on a Dipole in an Inhomogeneous Electric Field is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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