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Chapter 6: Problem 3

Write down the potential energy of a pair of charges, \(q\) at \(\boldsymbol{a}\) and \(-q\) at the origin, in a field with potential \(\phi(\boldsymbol{r})\). By considering the limit \(a \rightarrow 0\), show that the potential energy of a dipole of moment \(d\) is \(V=-\boldsymbol{d} \cdot \boldsymbol{E} .\) If the electric field is uniform, when is this potential energy a minimum? Show that the dipole experiences a net moment, or couple, \(\boldsymbol{G}=\boldsymbol{d} \wedge \boldsymbol{E}\), and that in a non-uniform field there is also a net force, \(\boldsymbol{F}=(\boldsymbol{d} \cdot \boldsymbol{\nabla}) \boldsymbol{E} .\) (Take \(\boldsymbol{d}\) in the \(z\)-direction, and show that \(\boldsymbol{F}=d \partial \boldsymbol{E} / \partial z .)\)

Short Answer

Expert verified
Answer: The minimum potential energy of a dipole in a uniform electric field occurs when the dipole moment is parallel to the electric field direction, and it is given by \(V = -\boldsymbol{d} \cdot \boldsymbol{E}\).

Step by step solution

01

Potential energy of the charge pair

First, let's write down the potential energy of a pair of charges. The potential energy V is given by: \(V = q \phi(\mathbf{a}) - q \phi(\mathbf{0})\)
02

Potential energy of a dipole

Now, consider the limit \(\mathbf{a}\) → 0. In this case, the dipole moment \(d = q\mathbf{a}\) holds. The potential energy V of a dipole can be found from: \(V=-\boldsymbol{d} \cdot \boldsymbol{E}\) where we also know that \(\boldsymbol{E}=-\boldsymbol{\nabla} \phi\)
03

Minimum potential energy

Consider the case when the electric field is uniform. In this case, the minimum potential energy V is the scalar projection of dipole moment \(\boldsymbol{d}\) onto the electric field \(\boldsymbol{E}\): \(V = -\boldsymbol{d} \cdot \boldsymbol{E}\) The potential energy is a minimum when the dipole moment is parallel to the electric field direction.
04

Net moment G

The net moment or couple \(\boldsymbol{G}\) experienced by the dipole is given by the vector product of the dipole moment and the electric field: \(\boldsymbol{G} = \boldsymbol{d} \wedge \boldsymbol{E}\)
05

Net force F in a non-uniform field

In a non-uniform electric field, the net force \(\boldsymbol{F}\) on the dipole can be written as: \(\boldsymbol{F}=(\boldsymbol{d} \cdot \boldsymbol{\nabla}) \boldsymbol{E}\) For the case when the dipole moment is in the z-direction, we have: \(\boldsymbol{F}= d \frac{\partial \boldsymbol{E}}{\partial z}\)

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