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Ch 26: Potential and Field
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 26: Potential and FieldProblema 79
Capitolo 26, Problema 79

Charge is uniformly distributed with charge density ρ inside a very long cylinder of radius R. Find the potential difference between the surface and the axis of the cylinder.

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Start by recognizing that the problem involves a uniformly distributed charge density (ρ) inside a long cylinder of radius R. Use Gauss's law to find the electric field inside the cylinder. For a Gaussian surface of radius r (where r < R), the enclosed charge is given by Q_enclosed = ρ * (πr²L), where L is the length of the cylinder.
Apply Gauss's law: ∮E·dA = Q_enclosed/ε₀. For the cylindrical Gaussian surface, the electric field E is constant over the surface, and the area of the curved surface is A = 2πrL. Substituting, E * (2πrL) = (ρ * πr²L)/ε₀. Simplify to find E = (ρ * r)/(2ε₀).
Now, calculate the potential difference between the axis (r = 0) and the surface (r = R). The potential difference is given by V = -∫E·dr, where the limits of integration are from r = 0 to r = R.
Substitute the expression for E into the integral: V = -∫[(ρ * r)/(2ε₀)] dr, with limits from 0 to R. This simplifies to V = -(ρ/(2ε₀)) ∫r dr, with the same limits.
Evaluate the integral: ∫r dr = (r²/2). Substituting the limits, the result is V = -(ρ/(2ε₀)) * [(R²/2) - (0²/2)]. Simplify this expression to find the potential difference in terms of ρ, R, and ε₀.

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Charge Density

Charge density (ρ) is a measure of the amount of electric charge per unit volume. In the context of a uniformly charged cylinder, it indicates how charge is distributed throughout the cylinder's volume. Understanding charge density is crucial for calculating electric fields and potentials in electrostatics.
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Intro to Density

Electric Potential

Electric potential is the work done per unit charge in bringing a test charge from infinity to a point in an electric field. It is a scalar quantity and is essential for determining the potential difference between two points, such as the surface and the axis of the cylinder in this problem.
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Electric Potential

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly useful for calculating electric fields in symmetric charge distributions, such as a long cylinder. By applying Gauss's Law, one can derive the electric field needed to find the potential difference in this scenario.
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Gauss' Law
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