Q2.45P - Introduction to Electrodynamics

Question

Find the electric field at a height $Z$ above the center of a square sheet (side a) carrying a uniform surface charge $\sigma$. Check your result for the limiting

cases $a\to \infty$ and $z>>a$.

Step-by-Step Solution

Answer

The electric filed is due to square plate is when $a\to \infty$ is $E=\frac{\sigma }{2{\epsilon }_0}$.

The electric field due to square plate $z>>a$ is $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{z^2}$.

a Step 1: Define functions

The expression for the electric filed at a distance $z$ above the center of the square loop carrying uniform line charge $\lambda$ is,

$E=\frac{1}{4\pi {\epsilon }_0}\frac{4\lambda az}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}\widehat{z}$

Here, $E$ is the electric filed, $\lambda$ is the linear charge density, ${\epsilon }_0$ is the permittivity for the free space, $a$is the length of each side of the square sheet.

The square sheet is shown in below figure.

Write the expression for linear charge density for the above square loop.

$d\lambda =\sigma \left(\frac{da}{2}\right)dE=\frac{1}{4\pi {\epsilon }_0}\frac{4az}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}d\lambda$

Here, $\sigma$ is the charge density.

Step 2: Determine linear charge density

Differentiating the equation $\left(1\right)$ on both sides,

Substitute $\sigma \left(\frac{da}{2}\right)$ for $d\lambda$.

width="233" style="max-width: none; vertical-align: -35px;" $\begin{array}{r}dE=\frac{1}{4\pi {\epsilon }_0}\frac{4az\sigma \left(\frac{da}{2}\right)}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}\end{array}$

$=\frac{1}{4\pi {\epsilon }_0}\left(\frac{4\sigma z}{2}\right)\frac{ada}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}$

$=\frac{\sigma z}{2\pi {\epsilon }_0}\frac{ada}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}$

Thus, the differential equation solution is $=\frac{\sigma z}{2\pi {\epsilon }_0}\frac{ada}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}$

Step 3: Determine Electric field

Now, integrate the equation $\left(1\right)$ with limits from $0$ to $a$ and solve for the electric filed due to the square sheet at a height z above its center.

$E=\frac{\sigma z}{2\pi {\epsilon }_0}{\int }_0^a \frac{a}{\left(z^2+\frac{a^2}{4}\right)\sqrt{z^2+\frac{a^2}{2}}}da$

Let $a^2=4t$, then $da=\frac{2dt}{a}$. The limits of $t$ are $0$ and $\frac{a^2}{4}$. Then the equation becomes,

$\begin{array}{r}E=\frac{\sigma z}{2\pi {\epsilon }_0}{\int }_0^{\frac{a^2}{4}} \frac{a}{\left(z^2+t\right)\sqrt{\left(z^2+2t\right)}}\frac{2}{a}dt\end{array}$

$=\frac{\sigma z}{\pi {\epsilon }_0}{\int }_0^{\frac{a^2}{4}} \frac{1}{\left(z^2+t\right)\sqrt{\left(z^2+2t\right)}}dt$

As, $\left(a^2+b\right)\sqrt{\left(a^2+2b\right)}={\tan }^{-1}\left(\frac{\sqrt{\left(a^2+2b\right)}}{a}\right)$

Solving further based on the above tangential formula,

$E=\frac{\sigma z}{\pi {\epsilon }_0}{\left[\frac{2}{z}{\tan }^{-1}\left(\frac{\sqrt{\left(z^2+2t\right)}}{z}\right)\right]}_0^{\frac{a^2}{4}}$
$=\frac{2\sigma }{\pi {\epsilon }_0}\left[{\tan }^{-1}\left(\frac{\sqrt{z^2+2\left(\frac{a^2}{4}\right)}}{z}\right)-{\tan }^{-1}\left(\frac{\sqrt{z^2+2\left(0\right)}}{z}\right)\right]$

$=\frac{2\sigma }{\pi {\epsilon }_0}\left[{\tan }^{-1}\left(\frac{\sqrt{z^2+\frac{a^2}{2}}}{z}\right)-{\tan }^{-1}\left(1\right)\right]$

$=\frac{2\sigma }{\pi {\epsilon }_0}\left[{\tan }^{-1}\left(\frac{\sqrt{z^2+\frac{a^2}{2}}}{z}\right)-{\tan }^{-1}\left(\tan \frac{\pi }{4}\right)\right]$

Simplifying the expression further,

$\begin{array}{r}E=\frac{2\sigma }{\pi {\epsilon }_0}\left[{\tan }^{-1}\left(\frac{\left.\left.\sqrt{z^2+\frac{a^2}{2}}\right)-\frac{\pi }{4}\right]}{z}\right]-1\right]\end{array}$

$=\frac{2\sigma }{\pi {\epsilon }_0}\frac{\pi }{4}\left[\left[\frac{4}{\pi }{\tan }^{-1}\right]\left(\frac{\sqrt{z^2+\frac{a^2}{2}}}{z}\right]-1\right]$

$=\frac{\sigma }{2\pi {\epsilon }_0}\left[\frac{4}{\pi }{\tan }^{-1}\left(\sqrt{1+\left(\frac{a^2}{2z^2}\right.}\right)-1\right]$

Thus, the electric filed is $\begin{array}{r}E=\frac{\sigma }{2\pi {\epsilon }_0}\left[\frac{4}{\pi }{\tan }^{-1}\left(\sqrt{1+\left(\frac{a^2}{2z^2}\right.}\right)-1\right]\end{array}$

If $a\to \infty$ then the electric field square plate is,

$E=\frac{2\sigma }{\pi {\epsilon }_0}\left[{\tan }^{-1}\left(\mathrm{\infty }\right)-\frac{\pi }{4}\right]$

Since, $\tan \frac{\pi }{2}=\infty$

$E=\frac{2\sigma }{\pi {\epsilon }_0}\left[{\tan }^{-1}\left(\tan \frac{\pi }{2}\right)-\frac{\pi }{4}\right]$

$=\frac{2\sigma }{\pi {\epsilon }_0}\left[\frac{\pi }{2}-\frac{\pi }{4}\right]$

$=\frac{\sigma }{2{\epsilon }_0}$

From the above equation, it is clear that the square sheet act as square plane. Thus the electric filed is due to square plate when $a\to \mathrm{\infty }$ is $E=\frac{\sigma }{2{\epsilon }_0}$

Step 4: Determine Electric field due to z >> a

Now, let’s consider that, $f\left(x\right)={\tan }^{-1}\sqrt{1+x}-\frac{x}{4}$ and $x=\frac{a^2}{2z^2}$. If $z>>a$, then $\frac{a^2}{2z^2}<<1$ and $f\left(0\right)=0$. Then the value of $f\text{'}\left(x\right)$ is,

$f\text{'}\left(x\right)=\frac{1}{1+\left(1+x\right)}\frac{1}{2}\frac{1}{\sqrt{1+x}}$

Then the value of $f\text{'}\left(0\right)$ by substituting $0$ for $x$ is,

$f\text{'}\left(0\right)=\frac{1}{4}$

Applying Taylor series and solving,

$f\left(x\right)=f\left(0\right)+xf\text{'}\left(0\right)+\frac{1}{2}{x}^{2}f\text{'}\text{'}\left(x\right)+...$

 $\approx f\left(0\right)+xf\text{'}\left(0\right)$

Substitute 0 for $f\left(0\right)$ and $\frac{1}{4}$ for $f\text{'}\left(0\right)$

$f\left(x\right)=0+\frac{x}{4}$

$=\frac{x}{4}$

Substitute $\frac{a^2}{2z^2}$ for $x$

$f\left(x\right)=\frac{a^2}{8z^2}$

Therefore, the electric filed due to square plate when $z>>a$ is,

$E=\frac{2\sigma }{\pi {\epsilon }_0}\left[\frac{a^2}{8z^2}\right]$

$=\frac{\sigma a^2}{4\pi {\epsilon }_0{z}^2}$

$=\frac{1}{4\pi {\epsilon }_0}\frac{q}{z^2}$

Thus from above result it is clear that, the square sheet acts as a point change when  $z>>a$.

Therefore, the electric field due to square plate $z>>a$ is $E=\frac{1}{4\pi {\epsilon }_0}\frac{q}{z^2}$.