91Ó°ÊÓ

An atom of polarizability $\mathrm{Î\pm }$ is kept in an electrostatic field $\stackrel{→}{\mathit{E}}$.

The force on an atom in an electrostatic field is

$\stackrel{\mathbf{→}}{\mathbf{F}}\mathbf{=}\mathbf{a}\mathbf{(}\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{.}\stackrel{\mathbf{→}}{\mathbf{∇}}\mathbf{)}\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Here, $\mathrm{Î\pm }$is polarizability and $\stackrel{→}{E}$is electrostatic field.

The gradient of square of the electrostatic field is

$\stackrel{→}{∇}{E}^2=2\stackrel{→}{E}×(\stackrel{→}{∇}×\stackrel{→}{E})+(\stackrel{→}{E}.\stackrel{→}{∇})\stackrel{→}{E}$

But the curl of an electrostatic field is zero. Hence

$\stackrel{→}{∇}{E}^2=2\left(\stackrel{→}{E}.\stackrel{→}{∇}\right)\stackrel{→}{E}$

Substitute this in equation (1) and get

$\stackrel{→}{F}=\frac{\mathrm{Î\pm }\stackrel{→}{∇}{E}^2}{2}$

Thus, the force on the atom is $\frac{1}{2}\mathrm{Î\pm }\stackrel{→}{∇}{E}^2$.

If $E^2$has a maxima, then there is a sphere about that maxima point $P$such that for all other points $P\text{'}≠P$ on the sphere

$\left|\stackrel{→}{E}(p\text{'})\right|<\left|\stackrel{→}{E}\left(p\right)\right|........\left(2\right)$

In the absence of any charge inside the sphere, the average field over the surface is equal to the field on the maxima, that is,

$\frac{1}{4{\mathrm{Ï€¸é}}^2}∫\stackrel{→}{E}da=\stackrel{→}{E}\left(P\right).......\left(3\right)$

Here, is the radius of the sphere and is the infinitesimal area element.

From equations (2) and (3) it follows

$$\begin{gathered} \frac{1}{4{\mathrm{Ï€¸é}}^2}∫\stackrel{→}{E}da<\frac{1}{4{\mathrm{Ï€¸é}}^2}∫\stackrel{→}{E}\left(p\right)da \\ \stackrel{→}{E}\left(p\right)<\frac{1}{4{\mathrm{Ï€¸é}}^2}∫\stackrel{→}{E}\left(p\right)da \\ \stackrel{→}{E}\left(p\right)<\stackrel{→}{E}\left(p\right) \end{gathered}$$

This is a contradiction and hence the electrostatic field does not have a maxima.