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Chapter 22: Problem 53

A \(-6.00-n C\) point charge is located at the center of a conducting spherical shell. The shell has an inner radius of \(2.00 \mathrm{~m},\) an outer radius of \(4.00 \mathrm{~m},\) and a charge of \(+7.00 \mathrm{nC}\) a) What is the electric field at \(r=1.00 \mathrm{~m} ?\) b) What is the electric field at \(r=3.00 \mathrm{~m} ?\) c) What is the electric field at \(r=5.00 \mathrm{~m} ?\) d) What is the surface charge distribution, \(\sigma,\) on the outside surface of the shell?

Short Answer

Expert verified
Answer: The electric field values and surface charge distribution are as follows: a) At r = 1m (inside the point charge): \(E_1 \approx 0 \, N/C\) b) At r = 3m (inside the spherical shell): \(E_2 \approx -11.623 \, N/C\) c) At r = 5m (outside the spherical shell): \(E_3 \approx 3.59 \times 10^{-3} \, N/C\) d) Surface charge distribution on the outside surface of the shell: \(\sigma \approx 1.39 \times 10^{-10} \, C/m^2\)

Step by step solution

01

Find the electric field at r = 1m (inside the point charge)

The electric field inside a point charge is uniformly zero. Therefore, the electric field at r = 1m is: \(E_1 = 0\)
02

Find the electric field at r = 3m (inside the spherical shell)

Applying Gauss's law for a Gaussian surface within the spherical shell (sphere of radius r = 3m), the electric field at r = 3m is: \(\oint \vec{E_2} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}\) \(q_{enclosed}=q\) (The point charge q is enclosed by the Gaussian surface) \(E_2 4 \pi (3)^2 = \frac{-6 \times 10^{-9}}{8.85 \times 10^{-12}}\) \(E_2 = \frac{-6 \times 10^{-9}}{8.85 \times 10^{-12} \times 36\pi}\) \(E_2 \approx -11.623 \, N/C\)
03

Find the electric field at r = 5m (outside the spherical shell)

Applying Gauss's law for a Gaussian surface outside the spherical shell (sphere of radius r = 5m), the electric field at r = 5m is: \(\oint \vec{E_3} \cdot d\vec{A} = \frac{Q_{total}}{\epsilon_0}\) \(q_{total}=q+Q\) \(E_3 4 \pi (5)^2 = \frac{(7 - 6) \times 10^{-9}}{8.85 \times 10^{-12}}\) \(E_3 = \frac{1 \times 10^{-9}}{8.85 \times 10^{-12} \times 100\pi}\) \(E_3 \approx 3.59 \times 10^{-3} \, N/C\)
04

Calculate the surface charge distribution on the outside surface of the shell

The shells charge Q is uniformly distributed on its outer surface. The surface charge density \(\sigma\) can be calculated as follows: \(\sigma = \frac{Q}{A}\) Here, \(A=4\pi b^2\) is the surface area of the outer surface of the spherical shell. \(\sigma = \frac{7 \times 10^{-9}}{4 \pi (4)^2}\) \(\sigma \approx 1.39 \times 10^{-10} \, C/m^2\) The electric field and surface charge distribution are as follows: a) \(E_1 \approx 0 \, N/C\) b) \(E_2 \approx -11.623 \, N/C\) c) \(E_3 \approx 3.59 \times 10^{-3} \, N/C\) d) \(\sigma \approx 1.39 \times 10^{-10} \, C/m^2\)

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Most popular questions from this chapter

Two parallel, uniformly charged, infinitely long wires carry opposite charges with a linear charge density \(\lambda=1.00 \mu \mathrm{C} / \mathrm{m}\) and are \(6.00 \mathrm{~cm}\) apart. What is the magnitude and direction of the electric field at a point midway between them and \(40.0 \mathrm{~cm}\) above the plane containing the two wires?

There is an electric field of magnitude \(150 .\) N/C, directed vertically downward, near the surface of the Earth. Find the acceleration (magnitude and direction) of an electron released near the Earth's surface.

A thin glass rod is bent into a semicircle of radius \(R\). A charge \(+Q\) is uniformly distributed along the upper half, and a charge \(-Q\) is uniformly distributed along the lower half as shown in the figure. Find the magnitude and direction of the electric field \(\vec{E}\) (in component form) at point \(P\), the center of the semicircle.

Which of the following statements is (are) true? a) There will be no change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed on the outer surface. b) There will be some change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed on the outer surface. c) There will be no change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed at the center of the sphere. d) There will be some change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed at the center of the sphere.

An infinitely long, solid cylinder of radius \(R=9.00 \mathrm{~cm},\) with a uniform charge per unit of volume of \(\rho=6.40 \cdot 10^{-8} \mathrm{C} / \mathrm{m}^{3},\) is centered about the \(y\) -axis. Find the magnitude of the electric field at a radius \(r=4.00 \mathrm{~cm}\) from the center of this cylinder.
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