A cylindrical charge has a length L and radius R. Denoting the charge density of the cylindrical charge by p, find the electric field at a distance r from the axis of the cylinder when
a)r<R and
b)r>R use Gauss'law
To find the electric field at a distance r from the axis of the cylindrical charge using Gauss' law, we can proceed as follows:
a) When r < R:
1. Begin by drawing a Gaussian surface in the form of a cylindrical Gaussian pillbox of radius r and height L. The height of the pillbox should be the same as that of the cylindrical charge.
2. Since the electric field in the radial direction is uniform, we can take a circular section of the Gaussian pillbox of radius r as our Gaussian surface.
3. Apply Gauss' law, which states that the electric flux through a closed surface is equal to the net charge enclosed by that surface divided by the permittivity of free space (ε₀). In this case, the flux through the Gaussian surface is equal to 2πrL times the electric field at a distance r from the axis.
4. The net charge enclosed within the Gaussian surface is given by Q_enc = p * πR²L, where p is the charge density of the cylindrical charge.
5. Set up the Gauss' law equation: 2πrL * E = Q_enc / ε₀.
6. Solve the equation for E, the electric field: E = (p * R²) / (2ε₀r).
b) When r > R:
1. Follow the same steps as above to set up the Gaussian surface in the form of a cylindrical Gaussian pillbox with a radius r and height L.
2. However, in this case, the net charge enclosed within the Gaussian surface is the total charge of the cylindrical charge, which is Q_enc = p * πR²L.
3. Apply Gauss' law: 2πrL * E = Q_enc / ε₀.
4. Solve the equation for E, the electric field: E = (p * R²) / (2ε₀r).
So, whether r is less than R or greater than R, the expression for the electric field at a distance r from the axis of the cylindrical charge is given by E = (p * R²) / (2ε₀r).