What is the net electric flux through the cylinder of FIGURE EX24.21?

검증된 단계별 안내

Identify the relevant law: The net electric flux through a closed surface is given by Gauss's Law, which states:

Φ = \frac{q (enclosed)}{ε}

, where

q

is the total charge enclosed by the surface and

ε

is the permittivity of free space.

Determine the charge enclosed by the cylinder: Examine the figure (FIGURE EX24.21) to identify the charges inside the cylindrical surface. Sum up all the charges enclosed by the cylinder to find

q

Substitute the value of the enclosed charge into Gauss's Law: Use the formula

Φ = \frac{q (enclosed)}{ε}

to calculate the net electric flux. Ensure that the charge is in Coulombs and the permittivity of free space is

8.85 \times 10^{-12} F \cdot m^{-1}

Account for symmetry: If the cylinder is symmetric and the electric field is uniform, the flux through the curved surface may cancel out, leaving only contributions from the ends of the cylinder. Verify this based on the figure.

Conclude the net flux: After substituting the values and considering symmetry, the net electric flux through the cylinder is determined by the total enclosed charge divided by the permittivity of free space. Ensure all units are consistent throughout the calculation.

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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 net electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it is expressed as Φ_E = Q_enc/ε_0, where Φ_E is the electric flux, Q_enc is the enclosed charge, and ε_0 is the permittivity of free space. This principle is fundamental for calculating electric fields in symmetrical charge distributions.

Electric Flux

Electric flux is a measure of the electric field passing through a given area. It is defined as the product of the electric field (E) and the area (A) through which it passes, adjusted for the angle (θ) between the field lines and the normal to the surface: Φ_E = E · A · cos(θ). Understanding electric flux is crucial for applying Gauss's Law effectively.

Symmetry in Charge Distributions

Symmetry in charge distributions simplifies the calculation of electric fields and flux. For example, in cylindrical or spherical geometries, the electric field can be assumed uniform over certain surfaces, allowing for straightforward application of Gauss's Law. Recognizing the symmetry helps in determining the enclosed charge and thus the net electric flux through the surface.