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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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 25: The Electric PotentialProblema 82a
Capitolo 25, Problema 82a

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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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.

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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.
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Intro to Density

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.
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Total Internal Reflection

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.
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