# Electrostatic Pressure Concept

There was a Question bothering me.

I tried solving it But couldn't So I finally went up to my teacher asked him for help . He told me that there was a formula for Electrostatic pressure $\rightarrow$

$$\mbox{Pressure}= \frac{\sigma^2}{2\epsilon_0}$$

And we had just to multiply it to the projected area = $\pi r^2$

When i asked him about the pressure thing he never replied.

So what is it actually.Can someone Derive it/Explain it please.

I haven't seen the term electrostatic pressure used explicitly before, but I can explain how to think about the problem.

You need to consider the total force on each hemisphere, which is of course the integral over the sphere of the (vector) force per unit area. Take, then, a surface element $dA$, with charge $\sigma dA$. As is nicely explained by Purcell, the force on such a surface element is given by the average of the electric field inside and outside. Since the field inside vanishes, the total force on the surface element is then $$d\mathbf{F}=\frac{1}{2}\sigma dA\times\frac{4\pi R^2\sigma}{4\pi\epsilon_0}\frac{\hat{\mathbf{r}}}{R^2}=\frac{\sigma^2}{2\epsilon_0}\hat{\mathbf{r}}\,dA.$$ By symmetry, the total force on each hemisphere will be along the axis of the problem, which I take in the $z$ direction. This total force will then be $$\mathbf{F}=\int d\mathbf{F}=\hat{\mathbf{z}}\int\frac{\sigma^2}{2\epsilon_0}\hat{\mathbf{z}}\cdot\hat{\mathbf{r}}dA=\hat{\mathbf{z}}\frac{\sigma^2}{2\epsilon_0}R^2\int\cos(\theta)d\Omega=\frac{\sigma^2\pi R^2}{2\epsilon_0}\hat{\mathbf{z}}.$$

The effect is indeed like having a gas inside exerting an outward pressure $p=\frac{dF}{dA}=\frac{\sigma^2}{2\epsilon_0}$, but this is hardly general - it depends on the precise, global arrangement of charges of this particular problem, while giving the impression of being a purely local thing (since it depends only on the "local" charge density, which is of course also a global parameter). If you do accept this "pressure" then yes, the total force is this constant pressure times the area vector of the surface, which is $\pi R^2\hat{\mathbf{z}}$.

• How did you integrate $\cos \theta$ with $d \Omega$ ? – Buraian Jan 20 at 16:10

I think you can do this by dimensional analysis. I'll do the calculation because your professor almost certainly won't accept it, so it's not cheating :-)

We have the three quantities 1/$\epsilon_0$, $\sigma^2$ and $R^n$, where we don't know $n$, and the product has to have the dimensions of force. The dimensions are:

$1/\epsilon_0 = L^3MT^{-4}A^{-2} = L^3MT^{-4}Q^{-2}T^2 = L^3MT^{-2}Q^{-2}$

$\sigma^2 = Q^2L^{-4}$

$R^n = L^n$

and of course force has dimensions $MLT^{-2}$. Multiply together $1/\epsilon_0$, $\sigma^2$ and $R^n$ and set the dimensions equal to $MLT^{-2}$ and you get:

$L^3MT^{-2}Q^{-2}Q^2L^{-4}L^n = MLT^{-2}$

which simplifies to:

$ML^{-1}L^nT^{-2} = MLT^{-2}$

and therefore $n = 2$ and the answer is A.

• Yes, this question was meant to be done by dimensional analysis . If they were asking for electrostatic pressure they could have asked for exact numerical prefactor. – Shubham Kumar Oct 4 '20 at 3:30

If you charge up a body, you get either an excess or deficiency of electrons on the body. It must be obvious to you that if we keep a bunch of similar charges close each charge will experience a repulsion. This repulsion per unit area of the body can be termed electrostatic pressure. To solve this problem I can give you three hints to solve this problem for which you won't be needing any calculus.

1) A charge cannot interact with its own electrostatic field.

2) Whenever you move across the surface of a conductor there is a discontinuity in the field around it.

3) Force divided by the surface normal to the force will give you the pressure on the surface.

Electrostatic pressure is the tension developed inside the sphere due to mutual repulsion among the charges of the same sphere. Its same like a rubber band is stretched from all the points outwards so a stress is developed in it

When charge is given to a conductor body then due to mutual repulsion between two charges on the two parts of the given conductor, a net force at a point on the surface of a charge conductor whose direction is normally outward. This mechanical force developed per unit area on the surface of charge conductor is also called electrostatic pressure/electrostatic stress.

Take the case of 3-D conductor and derive pressure exerted on a plate slightly separated from that conductor. First try to derive it on your own if not comfortable take help from uploaded picture.👍

• Use Mathjax for math writing – Young Kindaichi Nov 11 '20 at 15:41