A rod of length LL lies along the yy-axis with its center at the origin. The rod has a nonuniform linear charge density λ=a∣y∣ λ=a|y|, where a is a constant with the units C/m2. Draw a graph of λλ versus yy over the length of the rod.

Understand the problem: The rod lies along the y-axis, centered at the origin, and has a nonuniform linear charge density λ = a|y|. This means the charge density depends on the absolute value of the y-coordinate, increasing symmetrically as we move away from the origin in either direction along the rod. Identify the range of y: Since the rod has a length L and is centered at the origin, the y-coordinate ranges from -L/2 to +L/2. This will be the domain of the graph for λ versus y. Analyze the behavior of λ: The linear charge density λ = a|y| depends on the absolute value of y. For y \u003e 0, λ = ay, and for y \u003c 0, λ = a(-y). This results in a V-shaped graph symmetric about the origin. Sketch the graph: Plot λ on the vertical axis and y on the horizontal axis. For y = 0 (at the origin), λ = 0. As y increases from 0 to L/2, λ increases linearly as λ = ay. Similarly, as y decreases from 0 to -L/2, λ also increases linearly as λ = a(-y). The graph will have two straight lines meeting at the origin, forming a V-shape. Label the graph: Mark the endpoints of the graph at y = -L/2 and y = +L/2. At these points, the charge density is λ = a(L/2). Ensure the graph is symmetric about the origin and clearly shows the linear relationship between λ and |y|.

Ch 23: The Electric Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

All textbooks

Knight Calc 5th Edition

Ch 23: The Electric Field

Problem 68a

Chapter 23, Problem 68a

A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Draw a graph of $λ$ versus $y$ over the length of the rod.

Verified step by step guidance

Understand the problem: The rod lies along the y-axis, centered at the origin, and has a nonuniform linear charge density λ = a|y|. This means the charge density depends on the absolute value of the y-coordinate, increasing symmetrically as we move away from the origin in either direction along the rod.

Identify the range of y: Since the rod has a length L and is centered at the origin, the y-coordinate ranges from -L/2 to +L/2. This will be the domain of the graph for λ versus y.

Analyze the behavior of λ: The linear charge density λ = a|y| depends on the absolute value of y. For y > 0, λ = ay, and for y < 0, λ = a(-y). This results in a V-shaped graph symmetric about the origin.

Sketch the graph: Plot λ on the vertical axis and y on the horizontal axis. For y = 0 (at the origin), λ = 0. As y increases from 0 to L/2, λ increases linearly as λ = ay. Similarly, as y decreases from 0 to -L/2, λ also increases linearly as λ = a(-y). The graph will have two straight lines meeting at the origin, forming a V-shape.

Label the graph: Mark the endpoints of the graph at y = -L/2 and y = +L/2. At these points, the charge density is λ = a(L/2). Ensure the graph is symmetric about the origin and clearly shows the linear relationship between λ and |y|.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

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

Linear Charge Density

Linear charge density (λ) is defined as the amount of electric charge per unit length along a line. In this case, the charge density varies with the position along the rod, given by the equation λ = a|y|, where 'a' is a constant. Understanding this concept is crucial for analyzing how charge is distributed along the rod and how it affects the electric field around it.

Graphing Functions

Graphing functions involves plotting the relationship between two variables on a coordinate system. For this problem, you will graph λ as a function of y, which requires understanding how to represent the nonuniform charge density visually. This helps in visualizing how the charge density changes with position along the rod, which is essential for further calculations.

Electric Field Due to Charge Distribution

The electric field generated by a charge distribution is influenced by the amount and arrangement of charge. For a rod with a nonuniform charge density, the electric field at a point in space can be calculated by integrating the contributions from each infinitesimal segment of the rod. This concept is vital for understanding the implications of the charge distribution on the surrounding space.

Recommended video:

Guided course

06:28

Electric Field due to a Point Charge

Related Practice

Textbook Question

A rod of length L lies along the y-axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Find the electric field strength of the rod at distance x on the x-axis.

Textbook Question

A problem of practical interest is to make a beam of electrons turn a 90° corner. This can be done with the parallel-plate capacitor shown in FIGURE P23.55. An electron with kinetic energy 3.0×10⁻¹⁷ J enters through a small hole in the bottom plate of the capacitor. Should the bottom plate be charged positive or negative relative to the top plate if you want the electron to turn to the right? Explain.

views

Textbook Question

INT In a classical model of the hydrogen atom, the electron orbits the proton in a circular orbit of radius 0.053 nm. What is the orbital frequency in rev/s? The proton is so much more massive than the electron that you can assume the proton is at rest.

views

Textbook Question

An infinitely long sheet of charge of width L lies in the xy-plane between x = -L /2 and x = L /2. The surface charge density is h. Draw a graph of field strength E versus x for x > L /2.

Textbook Question

A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ=a|y|$ , where a is a constant with the units C/m². Determine the constant a in terms of $L$ and the rod's total charge $Q$ .

views

Textbook Question

An electric field can induce an electric dipole in a neutral atom or molecule by pushing the positive and negative charges in opposite directions. The dipole moment of an induced dipole is directly proportional to the electric field. That is, $\vec{p} = α \vec{E}$ , where α is called the polarizability of the molecule. A bigger field stretches the molecule farther and causes a larger dipole moment. An ion with charge q is distance r from a molecule with polarizability α. Find an expression for the force _{$\vec{E}$ ion on dipole}.

views