5 Engineering Example 8

5.1 Field strength on a cylinder

Problem in words

Given the electric field $\underset{̲}{E}$ on the surface of a cylinder, use Gauss’ law to find the charge per unit length.

Mathematical statement of problem

On the surface of a long cylinder of radius $a$ , the electric field is given by

$\underset{̲}{E}=\frac{{\rho }_L}{2\pi {\epsilon }_0}\frac{\left(a+bcos\theta \right)\underset{̲}{\stackrel{̂}{r}}-bsin\theta \underset{̲}{\theta }}{\left(a^2+2abcos\theta +b^2\right)}$

(using cylindrical polar co-ordinates) due to a line of charge a distance $b$ ( $<$ $a$ ) from the centre of the cylinder. Using Gauss’ law , find the charge per unit length.

Find the integral $\int \; <!--nolimits\; -->\int \; <!--nolimits\; -->\underset{̲}{E}\cdot \underset{̲}{dS}$ and by equating this to $\frac{Q}{{\epsilon }_0}$ , find $b$ in the expression for $\underset{̲}{E}$ , using the result

${\int \; <!--nolimits\; -->}_0^{2\pi }\frac{a+bcos\theta }{\left(a^2+2abcos\theta +b^2\right)}d\theta =\frac{2\pi }{a}$

Mathematical analysis

Consider a cylindrical section - as in the previous example, there are no contributions from the ends of the cylinder since the electric field has no normal component here. However, on the curved surface

$\underset{̲}{dS}=ad\theta dz\underset{̲}{\stackrel{̂}{r}}$

So

$\underset{̲}{E}\cdot \underset{̲}{dS}=\frac{{\rho }_L}{2\pi {\epsilon }_0}\frac{a+bcos\theta }{\left(a^2+2abcos\theta +b^2\right)}ad\theta dz$

Integrating over the curved surface of the cylinder

From Gauss’ law

$\frac{{\rho }_{L}l}{{\epsilon }_0}=\frac{Q}{{\epsilon }_0}$ so $Q={\rho }_{L}l$

Interpretation

Therefore the charge per unit length on the line of charge is given by ${\rho }_L$ (i.e. the charge per unit length is constant).

Task!

Verify Gauss’ theorem for the vector field $\underset{̲}{F}=x\underset{̲}{i}-y\underset{̲}{j}+z\underset{̲}{k}$ and the unit cube $0\le x\le 1$ , $0\le y\le 1$ , $0\le z\le 1$ .

1. Find the vector $\underset{̲}{\nabla }\cdot \underset{̲}{F}$ .
2. Evaluate the integral ${\int \; <!--nolimits\; -->}_{z=0}^1{\int \; <!--nolimits\; -->}_{y=0}^1{\int \; <!--nolimits\; -->}_{x=0}^1\underset{̲}{\nabla }\cdot \underset{̲}{F}dxdydz.$
3. For each side, evaluate the normal vector $\underset{̲}{dS}$ and the surface integral $\int \; <!--nolimits\; -->{\int \; <!--nolimits\; -->}_S\underset{̲}{F}\cdot \underset{̲}{dS}$ .
4. Show that the two sides of the statement of the theorem are equal.

Answer

Exercises

1. Verify Gauss’ theorem for the vector field $\underset{̲}{F}=4xz\underset{̲}{i}-y^2\underset{̲}{j}+yz\underset{̲}{k}$ and the cuboid $0\le x\le 2$ , $0\le y\le 3$ , $0\le z\le 4$ .
2. Verify Gauss’ theorem, using cylindrical polar coordinates, for the vector field $\underset{̲}{F}={\rho }^{-2}\underset{̲}{\stackrel{̂}{\rho }}$ over the cylinder $0\le \rho \le r_0$ , $-1\le z\le 1$ for
   1. $r_0=1$
   2. $r_0=2$
3. For $S$ being the surface of the tetrahedron with vertices at $\left(0,0,0\right)$ , $\left(1,0,0\right)$ , $\left(0,1,0\right)$ and $\left(0,0,1\right)$ , find the surface integral
   
   $\int \; <!--nolimits\; -->{\int \; <!--nolimits\; -->}_S\left(x\underset{̲}{i}+yz\underset{̲}{j}\right)\cdot \underset{̲}{dS}$
   1. directly
   2. by using Gauss’ theorem

   Hint :- When evaluating directly, show that the unit normal on the sloping face is $\frac{1}{\sqrt{3}}\left(\underset{̲}{i}+\underset{̲}{j}+\underset{̲}{k}\right)$ and that $\underset{̲}{dS}=\left(\underset{̲}{i}+\underset{̲}{j}+\underset{̲}{k}\right)dxdy$

Answer