A 2.0-cm-diameter parallel-plate capacitor with a spacing of 0.50 mm is charged to 200 V. What are (a) the total energy stored in the electric field and (b) the energy density?

검증된 단계별 안내

Step 1: Calculate the area of one plate of the capacitor. The diameter is given as 2.0 cm, so the radius is 1.0 cm (0.01 m). Use the formula for the area of a circle: A = πr².

Step 2: Determine the capacitance of the parallel-plate capacitor using the formula C = (ε₀A) / d, where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m), A is the area of the plates, and d is the separation between the plates (0.50 mm = 0.0005 m).

Step 3: Calculate the total energy stored in the capacitor using the formula U = (1/2)CV², where C is the capacitance and V is the voltage (200 V).

Step 4: Compute the energy density (u), which is the energy per unit volume, using the formula u = U / Volume. The volume of the capacitor is given by the product of the area of the plates and the separation distance: Volume = A × d.

Step 5: Combine the results from the previous steps to express the total energy stored (U) and the energy density (u) in terms of the given values and constants.

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주요 개념

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

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage. It is defined as C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the plates. For parallel-plate capacitors, capacitance can also be calculated using the formula C = ε₀(A/d), where ε₀ is the permittivity of free space, A is the area of the plates, and d is the separation between them.

Energy Stored in a Capacitor

The energy (U) stored in a capacitor is given by the formula U = 1/2 CV², where C is the capacitance and V is the voltage. This equation shows that the energy stored is proportional to the square of the voltage and the capacitance. Understanding this relationship is crucial for calculating the total energy stored in the electric field of the capacitor.

Energy Density

Energy density (u) in an electric field is defined as the energy stored per unit volume. For a capacitor, it can be calculated using the formula u = 1/2 εE², where ε is the permittivity of the material between the plates and E is the electric field strength. This concept helps in understanding how energy is distributed within the electric field of the capacitor.