Two 3.0-cm-diameter aluminum electrodes are spaced 0.50 mm apart. The electrodes are connected to a 100 V battery. What is the magnitude of the charge on each electrode?

검증된 단계별 안내

Step 1: Recognize that this is a problem involving a parallel-plate capacitor. The charge on each electrode can be determined using the formula for capacitance and the relationship between charge, capacitance, and voltage: \( Q = C \cdot V \).

Step 2: Calculate the capacitance \( C \) of the parallel-plate capacitor using the formula \( C = \frac{\varepsilon_0 \cdot A}{d} \), where \( \varepsilon_0 \) is the permittivity of free space \( (8.85 \times 10^{-12} \ \text{F/m}) \), \( A \) is the area of one electrode, and \( d \) is the separation between the plates.

Step 3: Compute the area \( A \) of one electrode using the formula for the area of a circle: \( A = \pi \cdot r^2 \), where \( r \) is the radius of the electrode. The diameter is given as 3.0 cm, so \( r = 1.5 \ \text{cm} = 0.015 \ \text{m} \).

Step 4: Substitute the values for \( \varepsilon_0 \), \( A \), and \( d \) into the capacitance formula \( C = \frac{\varepsilon_0 \cdot A}{d} \) to calculate the capacitance.

Step 5: Use the calculated capacitance \( C \) and the given voltage \( V = 100 \ \text{V} \) in the formula \( Q = C \cdot V \) to find the magnitude of the charge \( Q \) on each electrode.

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주요 개념

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

Capacitance

Capacitance is the ability of a system to store electric 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 electrodes. In this scenario, the capacitance can be calculated using the geometry of the electrodes and the distance between them.

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as E = V/d, where E is the electric field strength, V is the voltage, and d is the distance between the charges. Understanding the electric field is crucial for determining how charges interact in the given setup.

Charge Distribution

Charge distribution refers to how electric charge is spread over a surface or within a volume. In the case of the electrodes, the charge will distribute evenly across their surfaces due to their symmetry. The total charge on each electrode can be calculated once the capacitance is known, using the relationship Q = C * V.