FIGURE P31.38 shows the electric field inside a cylinder of radius $R equals 3.0$ mm. The field strength is increasing with time as $E = 1.0 \times 10^{8} t^{2}$ V/m, where t is in s. The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for $t is less than 0$ . Find an expression for the electric flux $Φ_{e}$ through the entire cylinder as a function of time.

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Step 1: Recall the formula for electric flux (Φₑ), which is defined as the integral of the electric field (E) over a surface area (A). For a uniform electric field inside the cylinder, the flux is given by Φₑ = E × A, where A is the cross-sectional area of the cylinder.

Step 2: Write the expression for the cross-sectional area of the cylinder. The area A is given by A = πR², where R is the radius of the cylinder. Substituting R = 3.0 mm = 3.0 × 10⁻³ m, we have A = π(3.0 × 10⁻³)².

Step 3: Substitute the given time-dependent electric field E = 1.0 × 10⁸t² V/m into the formula for electric flux. The flux becomes Φₑ = (1.0 × 10⁸t²) × A.

Step 4: Replace A with its expression in terms of R. The flux now becomes Φₑ = (1.0 × 10⁸t²) × π(3.0 × 10⁻³)².

Step 5: Simplify the expression for Φₑ to obtain the final time-dependent formula. The result will be Φₑ = (1.0 × 10⁸t²) × π × (9.0 × 10⁻⁶), which can be further simplified to Φₑ = 9.0π × 10²t².

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주요 개념

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

Electric Field

An electric field is a region around a charged object where other charged objects experience a force. It is quantified by the electric field strength (E), measured in volts per meter (V/m). In this scenario, the electric field inside the cylinder is given as a function of time, indicating that it changes as time progresses.

Electric Flux

Electric flux (Φₑ) is a measure of the quantity of electric field lines passing through a given area. It is calculated as the product of the electric field strength and the area through which it passes, adjusted for the angle between the field lines and the normal to the surface. The formula is Φₑ = E × A × cos(θ), where θ is the angle between the electric field and the normal to the surface.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space (ε₀). This principle is essential for calculating electric flux in scenarios with symmetrical charge distributions, such as the cylinder in this problem.