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Chapter 23: Problem 36

A very long insulating cylinder of charge of radius 2.50 \(\mathrm{cm}\) carries a uniform linear density of 15.0 \(\mathrm{nC} / \mathrm{m}\) . If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads 175 \(\mathrm{V} ?\)

Short Answer

Expert verified
The probe must be placed approximately 2.29 cm from the surface.

Step by step solution

01

Understand the Problem

We need to find the distance from the surface of a charged insulating cylinder where the electric potential difference relative to the surface is 175 V. The linear charge density is \( \lambda = 15 \, \text{nC/m} \) and the radius of the cylinder is \( R = 2.50 \, \text{cm} \).
02

Express Electric Potential Difference

The electric potential difference \( \Delta V \) between two points inside an infinite cylinder of charge is given by \( \Delta V = - \int \mathbf{E} \cdot \mathbf{dl} \), where \( \mathbf{E} \) is the electric field due to the cylinder. For points outside the cylinder, we use: \[ V(r) - V(R) = \frac{\lambda}{2 \pi \varepsilon_0} \ln \left( \frac{r}{R} \right) \] where \( \varepsilon_0 \) is the permittivity of free space.
03

Set Up the Equation

Plug in the given potential difference and solve for \( r \): \[ 175 = \frac{15 \times 10^{-9}}{2 \pi \times 8.85 \times 10^{-12}} \ln \left( \frac{r}{0.025} \right) \]
04

Simplify the Equation

Calculate the constant in front of the logarithm to simplify the expression: \[ 175 = \frac{15}{2 \pi \times 8.85} \times 10^{3} \ln \left( \frac{r}{0.025} \right) \] \[ \approx 269.6 \ln \left( \frac{r}{0.025} \right) \]
05

Solve for Distance r

Rearrange the equation to find \( r \): \[ \ln \left( \frac{r}{0.025} \right) = \frac{175}{269.6} \] \[ \ln \left( \frac{r}{0.025} \right) \approx 0.649 \] By exponentiating both sides, \[ \frac{r}{0.025} = e^{0.649} \] \[ r \approx 0.025 \times e^{0.649} = 0.025 \times 1.914 \approx 0.04785 \, \text{m} \]
06

Calculate Distance from Surface

Since \( r \) is the distance from the cylinder's axis and the voltmeter is placed at the surface (which is 0.025 m from the axis), the distance from the surface is: \[ d = r - 0.025 = 0.04785 - 0.025 \approx 0.02285 \, \text{m} \] or \( 2.29 \, \text{cm} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permittivity of Free Space
The permittivity of free space, denoted as \( \varepsilon_0 \), is a fundamental measure that describes how electric fields interact with the vacuum. Its value is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \, \text{m}^2 \).
This constant is crucial in formulas related to electromagnetism, especially when dealing with electric fields and potentials. In essence, it represents how much electric charge is required to generate an electric field in a vacuum.
The permittivity of free space is used in the formula for calculating electric potential difference, which reflects how electric potential changes in relation to charge and distance. It's an integral part of the equation that helps us understand interactions in a charge distribution such as around an insulating cylinder.
Linear Charge Density
Linear charge density, represented as \( \lambda \), measures the amount of electric charge per unit length of an object. It is often expressed in units like \( \text{nC/m} \) (nanocoulombs per meter).
In the given exercise, the cylinder has a linear charge density of 15.0 \( \text{nC/m} \), meaning there is 15 nanocoulombs of charge distributed uniformly along each meter of the cylinder.
This concept is vital when calculating the electric field and potential, as it determines the intensity and distribution of the electric charge around an object. When dealing with infinite or very long objects, linear charge density simplifies our calculations significantly.
Electric Field
The electric field, symbolized by \( \mathbf{E} \), represents the force per unit charge exerted on a small positive test charge placed in the field. Its units are newtons per coulomb (\( \text{N/C} \)).
An electric field explains how charge exerts influence on other charges in its surroundings. In our exercise, understanding the electric field created by the insulating cylinder is essential to determining the potential difference.
The electric field due to a line of charge like an insulating cylinder varies with distance from the line. This field is used in calculations to find the potential at different points, using the relationship \( \Delta V = - \int \mathbf{E} \cdot \mathbf{dl} \), which translates the field into a measurable potential difference.
Insulating Cylinder
An insulating cylinder is a cylindrical object that does not allow charge to move freely across it. Instead, it holds static charges in place. In the problem, we deal with a very long insulating cylinder, ensuring uniform charge distribution.
This setup allows us to assume symmetry in electric field distribution, simplifying the mathematical treatment. The radius and height of the cylinder become factors in calculating electric potential and field strength.
Understanding these properties is important because it influences how the potential is calculated and helps in integrating over areas efficiently. In our scenario, the radius was given as 2.50 cm and helped establish the boundary conditions for the problem.

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Most popular questions from this chapter

A point charge \(Q=+4.60 \mu \mathrm{C}\) is held fixed at the origin. A second point charge \(q=+1.20 \mu \mathrm{C}\) with mass of \(2.80 \times 10^{-4} \mathrm{kg}\) is placed on the \(x\) -axis, 0.250 \(\mathrm{m}\) from the origin. (a) What is the electric potential energy \(U\) of the pair of charges? (Take \(U\) to be zero when the charges have infinite scparation.) (b) The scond point charge is released from rest. What is its speed when its distance from the origin is (i) \(0.500 \mathrm{m} ;\) (ii) \(5.00 \mathrm{m} ;\) (iii) 50.0 \(\mathrm{m} ?\)

Two charges of equal magnitude \(Q\) are beld a distance \(d\) apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). and (b) Repeat part (a) for two charges having opposite signs.

Self-Energy of a Sphere of Charge. A solid sphere of radius \(R\) contains a total charge \(Q\) distributed uniformly through- out its volume. Find the energy needed to assemble this charge by bringing infinitesimal charges from far away. This energy is called the "self-energy" of the charge distribution. (Hint: After you have assembled a charge \(q\) in a sphere of radius \(r\) , how much energy would it take to add a spherical shell of thickness \(d r\) having charge \(d q ?\) Then integrate to get the total energy.)

(a) Calculate the potential energy of a system of two small spheres, one carrying a charge of 2.00\(\mu \mathrm{C}\) and the other a charge of \(-3.50 \mu \mathrm{C},\) with their centers separated by a distance of 0.250 \(\mathrm{m}\) . Assume zero potential energy when the charges are infinitely separated. (b) Suppose that one of the spheres is held in place and the other sphere, which has a mass of 1.50 \(\mathrm{g}\) , is shot away from it. What minimum initial speed would the moving sphere need in order to escape completely from the attraction of the fixed sphere? (To escape, the moving sphere would have to reach a velocity of zero when it was infinitely distant from the fixed sphere.)

A small particle has charge \(-5.00 \mu \mathrm{C}\) and mass \(2.00 \times 10^{-4} \mathrm{kg}\) . It moves from point \(A\) , where the electric potential is \(V_{A}=+200 \mathrm{V},\) to point \(B\) , where the electric potential is \(V_{B}=+800 \mathrm{V}\) . The electric force is the only force acting on the particle. The particle has speed 5.00 \(\mathrm{m} / \mathrm{s}\) at point \(A .\) What is its speed at point \(B ?\) Is it moving faster or slower at \(B\) than at \(A ?\) Explain.
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