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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.