6.3 Applying Gauss’s Law

Check that the electric fields for the sphere reduce to the correct values for a point charge.

A thin straight wire has a uniform linear charge density $\lambda_{0} .$ Find the electric field at a distance d from the wire, where d is much less than the length of the wire.

Would Gauss’s law be helpful for determining the electric field of two equal but opposite charges a fixed distance apart?

Discuss the restrictions on the Gaussian surface used to discuss planar symmetry. For example, is its length important? Does the cross-section have to be square? Must the end faces be on opposite sides of the sheet?

Recall that in the example of a uniform charged sphere, $\rho_{0} = Q / \left(\frac{4}{3} \pi R^{3}\right) .$ Rewrite the answers in terms of the total charge Q on the sphere.

A very long, thin wire has a uniform linear charge density of $50 \mu \text{C}/\text{m} .$ What is the electric field at a distance 2.0 cm from the wire?

Repeat your calculations for the preceding problem, given that the charge is distributed uniformly over the surface of a spherical conductor of radius 10.0 cm.

When a charge is placed on a metal sphere, it ends up in equilibrium at the outer surface. Use this information to determine the electric field of $+ 3.0 \mu \text{C}$ charge put on a 5.0-cm aluminum spherical ball at the following two points in space: (a) a point 1.0 cm from the center of the ball (an inside point) and (b) a point 10 cm from the center of the ball (an outside point).

Determine if approximate cylindrical symmetry holds for the following situations. State why or why not. (a) A 300-cm long copper rod of radius 1 cm is charged with +500 nC of charge and we seek electric field at a point 5 cm from the center of the rod. (b) A 10-cm long copper rod of radius 1 cm is charged with +500 nC of charge and we seek electric field at a point 5 cm from the center of the rod.…

The electric field at 2 cm from the center of long copper rod of radius 1 cm has a magnitude 3 N/C and directed outward from the axis of the rod. (a) How much charge per unit length exists on the copper rod? (b) What would be the electric flux through a cube of side 5 cm situated such that the rod passes through opposite sides of the cube perpendicularly?

Charge is distributed uniformly with a density $\rho$ throughout an infinitely long cylindrical volume of radius R. Show that the field of this charge distribution is directed radially with respect to the cylinder and that

$\begin{matrix} \\ \\ E = \frac{\rho r}{2 \epsilon_{0}} & & & \left(r \leq R\right) ; \\ E = \frac{\rho R^{2}}{2 \epsilon_{0} r} & & & \left(r \geq R\right) .\end{matrix}$

The electric field 10.0 cm from the surface of a copper ball of radius 5.0 cm is directed toward the ball’s center and has magnitude $4.0 \times 10^{2} \text{N}/\text{C} .$ How much charge is on the surface of the ball?

Charge is distributed throughout a spherical volume of radius R with a density $\rho = \alpha r^{2} ,$ where $\alpha$ is a constant. Determine the electric field due to the charge at points both inside and outside the sphere.

The volume charge density of a spherical charge distribution is given by $\rho \left(r\right) = \rho_{0} e^{− \alpha r} ,$ where $\rho_{0}$ and $\alpha$ are constants. What is the electric field produced by this charge distribution?