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Chapter 2: Problem 24

(a) Find the electric field on the \(z\) axis produced by an annular ring of uniform surface charge density \(\rho_{s}\) in free space. The ring occupies the region \(z=0, a \leq \rho \leq b, 0 \leq \phi \leq 2 \pi\) in cylindrical coordinates. \((b)\) From your part (a) result, obtain the field of an infinite uniform sheet charge by taking appropriate limits.

Short Answer

Expert verified
Answer: The electric field on the z-axis produced by an infinite uniform sheet charge is \(E_z = \frac{\rho_s}{2\epsilon_0}\), where \(\rho_s\) is the surface charge density and \(\epsilon_0\) is the vacuum permittivity.

Step by step solution

01

Set up the problem in cylindrical coordinates

We are given the ring occupies the region \(z=0, a \leq \rho \leq b, 0 \leq \phi \leq 2\pi\). So, we will first consider a small strip of width \(d\rho\) at a distance \(\rho\) from the \(z\) axis and at \(z=0\).
02

Find the electric field due to the small strip

Let's consider a point \(P\) on the \(z\) axis at a distance \(z\) from the origin. The electric field \(d\vec{E}\) due to the small strip at point \(P\) can be given by: $$d\vec{E} = \frac{\rho_s dA}{4\pi\epsilon_0} \frac{\vec{r}}{r^3}$$ Where \(\rho_s\) is the surface charge density, \(dA\) is the area of the small strip, \(\vec{r}\) is the position vector from the strip to point \(P\), and \(r\) is the distance between the strip and point \(P\). The area of the small strip can be given by \(dA = \rho d\rho d\phi\). The distance \(r\) can be given by: $$r = \sqrt{\rho^2 + z^2}$$
03

Find the total electric field

Now, we need to find the total electric field by integrating the electric field due to the small strip over the entire annular ring. Since the integration is over the entire ring, only the component of the electric field along the \(z\) direction will remain, as the radial components will cancel out. Therefore, the \(z\) component of the electric field can be given by: $$E_z = \frac{\rho_s}{4\pi\epsilon_0} \int_{a}^{b} \int_{0}^{2\pi} \frac{z}{(\rho^2 + z^2)^{3/2}} \rho d \rho d\phi$$
04

Solve the integral

To simplify the integral, introduce a variable \(u = \rho^2 + z^2\), and to change the order of integration like this: \(\int_{0}^{2\pi}d\phi \int_{a}^{b}\frac{z}{u^{3/2}}\rho d\rho\). Now we can solve it like this: $$E_z = \frac{\rho_s}{4\pi\epsilon_0} \int_{0}^{2\pi} d\phi \int_{a}^{b} \frac{z}{u^{3/2}} \rho d \rho$$ $$E_z = \frac{\rho_s z}{4\pi\epsilon_0} \int_{0}^{2\pi} d\phi \int_{a}^{b} \frac{\rho d\rho}{(\rho^2 + z^2)^{3/2}}$$ Now solving the integrals, we get: $$E_z = \frac{\rho_s z}{2\epsilon_0} \left[ \frac{b - a}{\sqrt{a^2 + z^2}\sqrt{b^2 + z^2}} \right]$$
05

Obtain the field of an infinite uniform sheet charge

To obtain the field of an infinite uniform sheet charge, we will take the limit \(a \to 0\) and \(b \to \infty\). In this limit, the electric field on the \(z\) axis becomes: $$E_z = \lim_{a \to 0, b \to \infty} \frac{\rho_s z}{2\epsilon_0} \left[ \frac{b - a}{\sqrt{a^2 + z^2}\sqrt{b^2 + z^2}} \right]$$ $$E_z = \frac{\rho_s}{2\epsilon_0}$$ So, the electric field produced by an infinite uniform sheet charge on the \(z\) axis is \(\frac{\rho_s}{2\epsilon_0}\).

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Most popular questions from this chapter

Point charges of \(50 \mathrm{nC}\) each are located at \(A(1,0,0), B(-1,0,0), C(0,1,0)\), and \(D(0,-1,0)\) in free space. Find the total force on the charge at \(A\).

Find \(\mathbf{E}\) at the origin if the following charge distributions are present in free space: point charge, \(12 \mathrm{nC}\), at \(P(2,0,6) ;\) uniform line charge density, \(3 \mathrm{nC} / \mathrm{m}\), at \(x=-2, y=3 ;\) uniform surface charge density, \(0.2 \mathrm{nC} / \mathrm{m}^{2}\) at \(x=2\).

A spherical volume having a \(2-\mu \mathrm{m}\) radius contains a uniform volume charge density of \(10^{15} \mathrm{C} / \mathrm{m}^{3}\). (a) What total charge is enclosed in the spherical volume? (b) Now assume that a large region contains one of these little spheres at every corner of a cubical grid \(3 \mathrm{~mm}\) on a side and that there is no charge between the spheres. What is the average volume charge density throughout this large region?

The electron beam in a certain cathode ray tube possesses cylindrical symmetry, and the charge density is represented by \(\rho_{v}=-0.1 /\left(\rho^{2}+10^{-8}\right)\) \(\mathrm{pC} / \mathrm{m}^{3}\) for \(0<\rho<3 \times 10^{-4} \mathrm{~m}\), and \(\rho_{v}=0\) for \(\rho>3 \times 10^{-4} \mathrm{~m} .(a)\) Find the total charge per meter along the length of the beam; \((b)\) if the electron velocity is \(5 \times 10^{7} \mathrm{~m} / \mathrm{s}\), and with one ampere defined as \(1 \mathrm{C} / \mathrm{s}\), find the beam current.

Two identical uniform sheet charges with \(\rho_{s}=100 \mathrm{nC} / \mathrm{m}^{2}\) are located in free space at \(z=\pm 2.0 \mathrm{~cm}\). What force per unit area does each sheet exert on the other?
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