An infinitely long cylindrical conductor has radius r r and uniform surface charge density σσ. In terms of σσ and RR, what is the charge per unit length λλ for the cylinder?

Ch 22: Gauss' Law

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 22: Gauss' Law

Problem 29a

Chapter 22, Problem 29a

An infinitely long cylindrical conductor has radius $r$ and uniform surface charge density $σ$ . In terms of $σ$ and $R$ , what is the charge per unit length $λ$ for the cylinder?

Verified step by step guidance

Understand the problem: We have an infinitely long cylindrical conductor with a radius r and a uniform surface charge density σ. We need to find the charge per unit length λ in terms of σ and R.

Recall the definition of surface charge density σ, which is the charge per unit area on the surface of the conductor. For a cylindrical surface, the area A of the curved surface is given by the formula: A = 2πrL, where L is the length of the cylinder.

Since the cylinder is infinitely long, we consider a finite length L for calculation purposes. The total charge Q on this length L of the cylinder can be expressed as: Q = σ * A = σ * 2πrL.

The charge per unit length λ is defined as the total charge Q divided by the length L of the cylinder. Therefore, λ = Q / L.

Substitute the expression for Q from step 3 into the formula for λ: λ = (σ * 2πrL) / L. Simplify this expression to find λ in terms of σ and r: λ = σ * 2πr.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Surface Charge Density

Surface charge density, denoted by σ, is the amount of electric charge per unit area on a surface. For a cylindrical conductor, it represents the charge distributed uniformly over the curved surface. Understanding σ is crucial for calculating the total charge on the cylinder's surface.

Charge per Unit Length

Charge per unit length, represented by λ, is the total charge distributed along a unit length of the cylinder. It is derived by integrating the surface charge density over the cylinder's surface area, providing a measure of how charge is distributed along the length of the conductor.

Cylindrical Geometry

Cylindrical geometry involves understanding the properties and dimensions of a cylinder, such as its radius and length. For an infinitely long cylinder, the focus is on the surface area and how charge is distributed over it, which is essential for calculating λ using the given σ and radius r.

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Textbook Question

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area $σ = 5.00 \times 10^{- 6}$ C/m². A small sphere of mass $m = 8.00 \times 10^{- 6}$ kg and charge $q$ is placed $3.00$ cm above the sheet of charge and then released from rest. If the sphere is to remain motionless when it is released, what must be the value of $q$ ?

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Textbook Question

A conductor with an inner cavity, like that shown in Fig. $22.23$ c, carries a total charge of $positive 5.00$ nC. The charge within the cavity, insulated from the conductor, is $negative 6.00$ nC. How much charge is on (a) the inner surface of the conductor and (b) the outer surface of the conductor?

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Textbook Question

An infinitely long cylindrical conductor has radius $r$ and uniform surface charge density $σ$ . In terms of $σ$ , what is the magnitude of the electric field produced by the charged cylinder at a distance $r > R$ from its axis? Then, express the result in terms of $λ$ and show that the electric field outside the cylinder is the same as if all the charge were on the axis.

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area $$\sigma$=5.00$\times$10^{-6}$ C/m². A small sphere of mass $m=8.00$\times$10^{-6}$ kg and charge $q$ is placed $3.00$ cm above the sheet of charge and then released from rest. What is $q$ if the sphere is released $1.50$ cm above the sheet?

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Textbook Question

A very long conducting tube (hollow cylinder) has inner radius $A$ and outer radius $b$ . It carries charge per unit length $+ α$ , where $α$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length. What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?

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Textbook Question

A very long conducting tube (hollow cylinder) has inner radius $A$ and outer radius $b$ . It carries charge per unit length $+α$ , where $α$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length. Calculate the electric field in terms of $α$ and the distance $r$ from the axis of the tube for (i) $r < a$ ; (ii) $a < r < b$ ; (iii) $r > b$ . Show your results in a graph of $E$ as a function of $R$ .

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