Question

To practice Problem-Solving Strategy 22.1: Gauss's Law.An infinite cylindrical rod has a uniform volume charge density rho(where rho>0). The cross section of the rod has radius r0. Find the magnitude of the electric field E at a distance r from the axis of the rod. Assume that r

Answer 1

The magnitude of the electric field at a distance r from the axis of an infinite cylindrical rod with uniform volume charge density ρ can be found using Gauss's law. The steps involve identifying cylindrical symmetry, choosing a co-axial cylindrical Gaussian surface, calculating the flux, and solving for E, which results in E = ρr/(2ε0) for r

Step-by-step explanation:

Applying Gauss's Law to Find the Electric Field

To find the magnitude of the electric field E at a distance r from the axis of an infinite cylindrical rod with a uniform volume charge density ρ, where r is less than the rod's radius r0, we will apply Gauss's law.

Steps to Use Gauss's Law

1. Identify the symmetry. Here, we have cylindrical symmetry.
2. Select an appropriate Gaussian surface. In this case, it's a cylinder co-axial with the rod and of radius r.
3. Calculate the electric flux through the Gaussian surface. The electric field E is constant over the surface, and perpendicular to it, thus simplifying the integral to E⋅area.
4. Equate the electric flux to the charge enclosed divided by the permittivity of free space (ε0), following Gauss's law ΦE = Qenc /ε0.
5. Solve for the electric field E.

In this cylindrical symmetry, the enclosed charge Qenc is equal to the volume charge density ρ multiplied by the volume of the cylinder within radius r (given by ρ⋅π⋅r2⋅L, where L is the length of the Gaussian cylinder which cancels out eventually).

Solution

By Gauss's law, E(2πrL) = ρπr2L/ε0, where E is the electric field, r is the distance from the axis, and L is the length of the Gaussian surface. Solving for E, we find E = ρr/(2ε0). This formula gives the electric field at a point within the cylindrical rod's radius.

Answer 2

Solution for 3 different cases are given in explanation.

Step-by-step explanation:

Gauss's Law:

`\int_S E.dA=Q_(encl)/\epsilon_0`

for
`r<r_0` :

`E.2\pi rL=(\rho\pi r^2L/\pi r_0^2)/(\epsilon_0) \\\\E=(\rho r)/(2\pi\epsilon_0 r_0^2)`

for
`r=r_0` :

`E.2\pi r_0L=(\rho\pi r_0^2L)/(\epsilon_0) \\\\E=(\rho r_0)/(2\epsilon_0)`

for
`r>r_0` :

`E.2\pi rL=(\rho\pi r_0^2L)/(\epsilon_0) \\\\E=(\rho r_0^2)/(2\epsilon_0 r)`