Chapter 5.3, Problem 3aT

A large sheet has charge density $+{\sigma }_o$ . A cylindrical Gaussian surface encloses a portion of the sheet andextends a distance $L_o$ , on either side of the sheet. $A_1,A_2,\text{and}{A}_3$ are the areas of the ends and curved side, respectively. Only a small portion of the sheet is shown.

1. On the diagram at right indicate the location of the charge enclosed by the Gaussian cylinder.

In terms of ${\sigma }_o$ , and other relevant quantities, what is the net charge enclosed by the Gaussian cylinder?

2. Sketch the electric field lines on both sides of the sheet.

Does the Gaussian cylinder affect the field lines or the charge distribution? Explain.

3. Let $E_{1}and{E}_R$ represent the magnitude of the electric field on the left and right ends of the Gaussian surface.

How do the magnitudes of $E_{\text{L}}and{E}_{\text{R}}$ compare? Explain.

How do the magnitudes of the areas of the ends of the Gaussian surface compare?

4. Through which of the surfaces $\left(A_1,A_2,A_3\right)$ is there a net flux? Explain using a sketch showing the relative orientation of the electric field vector and the area vectors.

Write an expression for the net electric flux ${\varphi }_{\text{net}}$ through the cylinder in terms of the three areas $\left(A_1,A_2,\text{and}{A}_3\right)E_{\text{L}},\text{and}{E}_{\text{R}}$ .

Use the relationships between the electric fields $E_{\text{L}}\text{and}{E}_{\text{R}}$ and between the areas $A_1$ , and $A_2$ to simplify your equation for the net flux.

5. Gauss’ law $\left({\varphi }_{\text{E}}=q_{\text{enclosed}}/{\epsilon }_o\right)$ relates the net electric flux through a gaussian surface (which you found in part 4) to the net charge enclosed (which you found in part 1). Use this relationship to find the direction and magnitude of the electric field at the right and of the cylinder in terms of ${\sigma }_o$ .

What is the electric field at the left end of the cylinder?

Does the electric field near a large sheet of charge depend on the distance from the sheet? Use your results above to justify your answer.

Is your answer consistent with the electric field lines you sketched in part 2?