A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ0x/Lλ(x)=λ_0x/L, where x is measured from the rod's left end. What is λ0λ_0 in terms of Q and L?

Ch 25: The Electric Potential

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 25: The Electric Potential

Problem 82a

Chapter 25, Problem 82a

A thin rod of length L and total charge Q has the nonuniform linear charge distribution $λ (x) = λ_{0} x / L$ , where x is measured from the rod's left end. What is $lamda sub 0$ in terms of Q and L?

Verified step by step guidance

The linear charge density λ(x) is given as λ(x) = λ₀x/L, where λ₀ is a constant to be determined. The total charge Q on the rod is the integral of the linear charge density over the length of the rod.

Set up the integral for the total charge Q: Q = ∫[0 to L] λ(x) dx. Substituting λ(x) = λ₀x/L, the integral becomes Q = ∫[0 to L] (λ₀x/L) dx.

Factor out constants from the integral: Q = (λ₀/L) ∫[0 to L] x dx.

Evaluate the integral of x with respect to x: ∫[0 to L] x dx = [x²/2] evaluated from 0 to L. This simplifies to (L²/2).

Substitute the result of the integral back into the equation for Q: Q = (λ₀/L) * (L²/2). Simplify to solve for λ₀: λ₀ = 2Q/L².

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 charged object. In this case, the charge distribution is nonuniform, meaning it varies along the length of the rod. The expression λ(x) = λ₀x/L indicates that the charge density increases linearly from the left end to the right end of the rod.

Total Charge

Total charge (Q) is the sum of all electric charges present in a system. For the rod with a nonuniform charge distribution, the total charge can be calculated by integrating the linear charge density over the length of the rod. This relationship is crucial for determining λ₀, as it connects the charge distribution to the overall charge of the rod.

Integration in Physics

Integration is a mathematical tool used to find the total quantity from a variable distribution. In this context, it allows us to calculate the total charge Q by integrating the linear charge density λ(x) over the length of the rod. This process is essential for deriving relationships between the charge density and the total charge, particularly when dealing with nonuniform distributions.

Recommended video:

Guided course

11:43

Finding Moment Of Inertia By Integrating

Related Practice

Textbook Question

An electric dipole consists of 1.0 g spheres charged to ±2.0 nC at the ends of a 10-cm-long massless rod. The dipole rotates on a frictionless pivot at its center. The dipole is held perpendicular to a uniform electric field with field strength 1000 V/m, then released. What is the dipole's angular velocity at the instant it is aligned with the electric field?

views

Textbook Question

Two 2.0-mm-diameter beads, C and D, are 10 mm apart, measured between their centers. Bead C has mass 1.0 g and charge 2.0 nC. Bead D has mass 2.0 g and charge −1.0 nC. If the beads are released from rest, what are the speeds v_C and v_D at the instant the beads collide?

views

Textbook Question

A hollow cylindrical shell of length L and radius R has charge Q uniformly distributed along its length. What is the electric potential at the center of the cylinder?

views

Textbook Question

A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ₀x/L, where x is measured from the rod's left end. What is the electric potential on the axis at distance d left of the rod's left end?

views

Textbook Question

Bead A has a mass of 15 g and a charge of −5.0 nC. Bead B has a mass of 25 g and a charge of −10.0 nC. The beads are held 12 cm apart (measured between their centers) and released. What maximum speed is achieved by each bead?

views