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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 textbooksKnight Calc 5th EditionCh 25: The Electric PotentialProblem 82b
Chapter 25, Problem 82b

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?

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Understand the problem: The electric potential at a point due to a charge distribution is calculated by integrating the contributions from each infinitesimal charge element. Here, the charge distribution is nonuniform, given by λ(x) = λ₀x/L, and we need to find the potential at a point located at a distance d to the left of the rod's left end.
Express the infinitesimal charge element: The linear charge density λ(x) is the charge per unit length. For a small segment of the rod of length dx at position x, the infinitesimal charge is dQ = λ(x)dx. Substituting λ(x), we get dQ = (λ₀x/L)dx.
Write the expression for the electric potential: The electric potential dV at a point due to an infinitesimal charge dQ is given by dV = (1 / (4πε₀)) * (dQ / r), where r is the distance from the charge element to the point of interest. Here, r = d + x, so dV = (1 / (4πε₀)) * ((λ₀x/L)dx / (d + x)).
Set up the integral: To find the total electric potential V, integrate dV over the length of the rod. The limits of integration are from x = 0 (left end of the rod) to x = L (right end of the rod). Thus, V = ∫[0 to L] (1 / (4πε₀)) * ((λ₀x/L) / (d + x)) dx.
Simplify and solve the integral: Factor out constants (1 / (4πε₀)) * (λ₀ / L) from the integral. The remaining integral is ∫[0 to L] (x / (d + x)) dx. Use substitution or partial fraction techniques to evaluate this integral. Once the integral is solved, substitute the limits to find the expression for the electric potential V.

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Key Concepts

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

Electric Potential

Electric potential is the amount of electric potential energy per unit charge at a point in an electric field. It is a scalar quantity measured in volts (V) and indicates the work done to move a unit positive charge from a reference point to a specific point in the field without any acceleration.
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Electric Potential

Linear Charge Density

Linear charge density (λ) is defined as the amount of electric charge per unit length along a line, typically expressed in coulombs per meter (C/m). In this case, the charge distribution is nonuniform, meaning it varies along the length of the rod, which affects the calculation of the electric potential at different points.
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Integration in Electric Field Calculations

In problems involving continuous charge distributions, integration is used to calculate quantities like electric potential. This involves summing contributions from infinitesimal charge elements along the distribution, taking into account their distances from the point of interest, which allows for the determination of the total potential at a specific location.
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Related Practice
Textbook Question

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?

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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 vC and vD at the instant the beads collide?

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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?

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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?

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