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Chapter 22: Problem 33

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.

Short Answer

Expert verified
Answer: The electric field at the center of the semicircle is \(\vec{E} = -\dfrac{2kQ}{R}\hat{j}\), where \(k\) is the Coulomb constant, \(Q\) is the total charge on each half of the semicircle, and \(R\) is the radius of the semicircle.

Step by step solution

01

Define a small charge element on the semicircle

Let's define a small charge element \(dq\) at an angle \(\theta\) on the semicircle. For the positive charges on the upper half, the angle \(\theta\) goes from \(0\) to \(\pi\) radians. For the negative charges on the lower half, the angle \(\theta\) goes from \(\pi\) to \(2\pi\) radians.
02

Determine the electric field contribution due to a small charge element

The electric field contribution from a small charge element \(dq\) is given by \(d\vec{E} = k \dfrac{dq}{r^2}\hat{r}\), where \(k\) is the Coulomb constant, \(r = R\) (distance from charge element \(dq\) to point \(P\)), and \(\hat{r}\) is the unit vector pointing from \(dq\) to \(P\).
03

Determine the components of the electric field due to the small charge element

To determine the \(X\) and \(Y\) components of the electric field, let's express \(d\vec{E}\) in component form: \(d\vec{E}_x = d\vec{E}\cos\theta = k \dfrac{dq}{R^2}\cos\theta\hat{i}\) \(d\vec{E}_y = d\vec{E}\sin\theta = k \dfrac{dq}{R^2}\sin\theta\hat{j}\)
04

Integrate the electric field components over the entire semicircle

Now we need to integrate these components over the entire semicircle. We begin with the positive charges on the upper half \((0 \leq \theta \leq \pi)\): \(\vec{E}_{+x} = \int_{0}^{\pi} k \dfrac{dq}{R^2}\cos\theta\hat{i}\) \(\vec{E}_{+y} = \int_{0}^{\pi} k \dfrac{dq}{R^2}\sin\theta\hat{j}\) Then, we perform the same integration for the negative charges on the lower half \((\pi \leq \theta \leq 2\pi)\): \(\vec{E}_{-x} = \int_{\pi}^{2\pi} k \dfrac{dq}{R^2}\cos\theta\hat{i}\) \(\vec{E}_{-y} = \int_{\pi}^{2\pi} k \dfrac{dq}{R^2}\sin\theta\hat{j}\)
05

Add the contributions from the positive and negative charges

To find the total electric field at point \(P\), we add the contributions from the positive and negative charges: \(\vec{E}_{x} = \vec{E}_{+x} + \vec{E}_{-x}\) \(\vec{E}_{y} = \vec{E}_{+y} + \vec{E}_{-y}\)
06

Find the final expression for the electric field

After completing the integration, we find that the electric field components are: \(\vec{E}_{x} = 0\hat{i}\) \(\vec{E}_{y} = -\dfrac{2kQ}{R}\hat{j}\) Thus, the total electric field at point \(P\) is: \(\vec{E} = \vec{E}_{x} + \vec{E}_{y} = -\dfrac{2kQ}{R}\hat{j}\)

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Most popular questions from this chapter

An electric dipole consists of two equal and opposite charges situated a very small distance from each other. When the dipole is placed in a uniform electric field, which of the following statements is true? a) The dipole will not experience any net force from the electric field; since the charges are equal and have opposite signs, the individual effects will cancel out. b) There will be no net force and no net torque acting on the dipole. c) There will be a net force but no net torque acting on the dipole. d) There will be no net force, but there will (in general) be a net torque acting on dipole.

Three \(-9-\mathrm{mC}\) point charges are located at (0,0) \((3 \mathrm{~m}, 3 \mathrm{~m})\), and \((3 \mathrm{~m},-3 \mathrm{~m})\). What is the magnitude of the electric field at \((3 \mathrm{~m}, 0) ?\) a) \(0.9 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) b) \(1.2 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) c) \(1.8 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) d) \(2.4 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) e) \(3.6 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) f) \(5.4 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) g) \(10.8 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\)

The electric flux through a spherical Gaussian surface of radius \(R\) centered on a charge \(Q\) is \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right) .\) What is the electric flux through a cubic Gaussian surface of side \(R\) centered on the same charge \(Q ?\) a) less than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) b) more than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) c) equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) d) cannot be determined from the information given

Consider a uniform nonconducting sphere with a charge \(\rho=3.57 \cdot 10^{-6} \mathrm{C} / \mathrm{m}^{3}\) and a radius \(R=1.72 \mathrm{~m}\). What is the magnitude of the electric field \(0.530 \mathrm{~m}\) from the center of the sphere?

A point charge, \(q=4.00 \cdot 10^{-9} \mathrm{C},\) is placed on the \(x\) -axis at the origin. What is the electric field produced at \(x=25.0 \mathrm{~cm} ?\)
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