A 20-cm-radius ball is uniformly charged to 80 nC. How much charge is enclosed by spheres of radii 5, 10, and 20 cm?

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Understand the problem: The ball is uniformly charged, meaning the charge is distributed evenly throughout its volume. To determine the charge enclosed by spheres of different radii, we need to use the concept of charge density and the relationship between the volume of a sphere and its radius.

Calculate the charge density (ρ): The charge density is the total charge divided by the total volume of the ball. The volume of a sphere is given by the formula:

V_{total} = \frac{4}{3} π R^{3}

, where R is the radius of the ball (20 cm). Use this to find

ρ = \frac{Q}{V_{total}}

, where Q is the total charge (80 nC).

Relate the charge enclosed to the radius of the smaller spheres: For a sphere of radius r (where r ≤ R), the charge enclosed is proportional to the volume of that sphere. The volume of a smaller sphere is given by:

V_{enclosed} = \frac{4}{3} π r^{3}

. The charge enclosed is then:

Q_{enclosed} = ρ V_{enclosed}

Substitute the values for each radius: For r = 5 cm, 10 cm, and 20 cm, calculate the enclosed volume and then multiply by the charge density to find the enclosed charge. Use the formula:

Q_{enclosed} = \frac{Q}{V_{total}} V_{enclosed}

Interpret the results: For r = 20 cm, the enclosed charge should equal the total charge (80 nC), since the sphere encompasses the entire ball. For r = 5 cm and 10 cm, the enclosed charge will be smaller, proportional to the cube of the radius. Ensure the results make sense based on the uniform charge distribution.

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주요 개념

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

Gauss's Law

Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it is expressed as Φ_E = Q_enc/ε₀, where Φ_E is the electric flux, Q_enc is the enclosed charge, and ε₀ is the permittivity of free space. This principle is fundamental for analyzing electric fields around charged objects.

Electric Field of a Charged Sphere

For a uniformly charged sphere, the electric field outside the sphere behaves as if all the charge were concentrated at the center. Inside the sphere, the electric field is zero. This concept is crucial for determining the electric field at various points relative to the charged sphere, particularly when applying Gauss's Law.

Charge Distribution

Charge distribution refers to how electric charge is spread over a given volume or surface. In this scenario, the ball is uniformly charged, meaning the charge is evenly distributed across its surface. Understanding charge distribution is essential for calculating the total charge enclosed within different spherical radii.