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

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 68b

Chapter 23, Problem 68b

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 $Q$ .

Verified step by step guidance

Step 1: Understand the problem. The rod has a nonuniform linear charge density λ = a|y|, where λ represents the charge per unit length at a given point y along the rod. The total charge Q is distributed along the rod of length L, and we need to find the constant 'a' in terms of Q and L.

Step 2: Express the total charge Q as the integral of the linear charge density λ over the length of the rod. Since the rod lies along the y-axis and is centered at the origin, its endpoints are at y = -L/2 and y = L/2. The total charge is given by: Q = ∫(λ dy) = ∫(a|y| dy), where the integral is evaluated from y = -L/2 to y = L/2.

Step 3: Break the integral into two parts to account for the absolute value of y. For y < 0, |y| = -y, and for y > 0, |y| = y. Therefore, the integral becomes: Q = ∫(a(-y) dy) from y = -L/2 to y = 0 + ∫(a(y) dy) from y = 0 to y = L/2.

Step 4: Solve each integral separately. For the first integral (from y = -L/2 to y = 0), substitute |y| = -y, and for the second integral (from y = 0 to y = L/2), substitute |y| = y. Combine the results to express Q in terms of 'a' and 'L'.

Step 5: Rearrange the equation to solve for 'a'. Since Q is known, isolate 'a' by dividing both sides of the equation by the appropriate terms involving L. This will give the expression for 'a' in terms of Q and L.

Verified video answer for a similar problem:

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

Video duration:

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 λ = a|y|, indicating that the charge density increases linearly with the distance from the center of the rod. Understanding this concept is crucial for calculating the total charge distributed along the length of the rod.

Total Charge Calculation

The total charge (Q) of the rod can be calculated by integrating the linear charge density over the length of the rod. This involves setting up an integral from -L/2 to L/2, where the charge density λ varies with y. The result of this integration will provide a relationship between the constant a, the length L, and the total charge Q, which is essential for solving the problem.

Integration in Physics

Integration is a fundamental mathematical tool used in physics to find quantities that accumulate over a continuous range, such as charge, mass, or energy. In this context, it allows us to sum the contributions of the varying charge density along the length of the rod. Mastery of integration techniques is necessary to derive the total charge from the given linear charge density function.

Recommended video:

Guided course

11:43

Finding Moment Of Inertia By Integrating

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

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. Draw a graph of $λ$ versus $y$ over the length of the rod.

views

Textbook Question

One type of ink-jet printer, called an electrostatic ink-jet printer, forms the letters by using deflecting electrodes to steer charged ink drops up and down vertically as the ink jet sweeps horizontally across the page. The ink jet forms 30-μm-diameter drops of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page at a speed of 20 m/s . Along the way, the drops pass through two horizontal, parallel electrodes that are 6.0 mm long, 4.0 mm wide, and spaced 1.0 mm apart. The distance from the center of the electrodes to the paper is 2.0 cm. To form the tallest letters, which have a height of 6.0 mm, the drops need to be deflected upward (or downward) by 3.0 mm. What electric field strength is needed between the electrodes to achieve this deflection? Ink, which consists of dye particles suspended in alcohol, has a density of 800 kg/m³.

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