Answer to Question #135892 in Electricity and Magnetism for rainy season

Question

State the laws of electrostatics in integral form.
(b) A uniformly charged long cylinder of radius a and length L has
total charge q inside its volume.
What is the direction of the electric field at points outside the cylinder?
Find the electric field inside and outside the cylinder.

Answer 1

(a) $\displaystyle \oint_S \bar{E}\cdot \bar{dS}=\frac{q}{\epsilon_0}$ (Gauss law in SI)

$\displaystyle \oint_C \bar{E}\cdot \bar{dl}=0$ (conservative filed law)

(b) Due to cylindrical symmetry, the electric field will be pointed from the axis of symmetry to outside. Let's use Gauss law.

(i) inside the cylinder (r<a)

$\displaystyle \int_0^a \rho(r) dV =q$ (in volume $\pi a^2L$ )

$\displaystyle E \cdot 2\pi r L = \frac{1}{\epsilon_0} \int_0^r \rho(r) dV = \frac{1}{\epsilon_0} q \frac{\pi r^2 L}{\pi a^2 L}= \frac{1}{\epsilon_0} q \frac{r^2}{a^2}$

$\displaystyle E = \frac{qr}{\epsilon_0 \, 2 \pi L a^2}$

(ii) outside of the cylinder (r>a)

$\displaystyle E \cdot 2\pi r L = \frac{1}{\epsilon_0} \int_0^a \rho(r) dV = \frac{1}{\epsilon_0} q$

$\displaystyle E = \frac{q}{\epsilon_0 \,2 \pi L r}$