Question

From Gauss's law, the electric field set up by a uniform line of charge is given by the following expression where is a unit vector pointing radially away from the line and λ is the linear charge density along the line. = Derive an expression for the potential difference between r = r1 and r = r2. (Use any variable or symbol stated above along with the following as necessary: ε0 and π.)

Answer 1

Answer:

$\Delta V=\lambda *ln(r_{2}/r_{1}) /\ (2\pi*\epsilon_{o})$

Explanation:

Using the Gauss Law, we obtain the electric Field for a uniform large line of charge:

$2\pi r L*E=\lambda *L/\epsilon_{o}$

$E=\lambda /\(2 \pi* r *\epsilon_{o})$

We calculate the potential difference from the electric field:

$\Delta V=-\int\limits^{r_{1}}_{r_{2}} E \, dr =-\int\limits^{r_{1}}_{r_{2}} \lambda dr/ (2\pi*r*\epsilon_{o})=\lambda *ln(r_{2}/r_{1}) /\ (2\pi*\epsilon_{o})$