(II) A 25-mH coil whose resistance is 0.80 Ω is connected to a capacitor C and a 420-Hz source voltage. If the current and voltage are to be in phase, what value must C have?

검증된 단계별 안내

Step 1: Understand that for the current and voltage to be in phase in an RLC circuit, the inductive reactance (XL) must equal the capacitive reactance (XC). The formula for inductive reactance is XL = 2\(\pi\) f L, where f is the frequency and L is the inductance.

Step 2: Calculate the inductive reactance using the given values: frequency (f = 420 Hz) and inductance (L = 25 mH = 0.025 H).

Step 3: Use the formula for capacitive reactance, which is XC = 1 / (2\(\pi\) f C), where C is the capacitance that needs to be calculated.

Step 4: Set the inductive reactance equal to the capacitive reactance (XL = XC) and solve for the capacitance C.

Step 5: Rearrange the equation to solve for C, which will be C = 1 / (2\(\pi\) f XL). Substitute the value of XL calculated in Step 2 to find the required capacitance.

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주요 개념

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

Inductance and Resistance in RLC Circuits

In an RLC circuit, the inductance (L) and resistance (R) determine how the circuit responds to alternating current (AC). The inductance creates a magnetic field that opposes changes in current, while resistance dissipates energy. The relationship between these components affects the phase difference between voltage and current, which is crucial for understanding how to achieve in-phase conditions.

Capacitance and Phase Relationship

Capacitance (C) is the ability of a capacitor to store charge. In an AC circuit, the capacitor causes the current to lead the voltage by 90 degrees. To achieve a condition where the current and voltage are in phase, the capacitive reactance must equal the inductive reactance, which can be calculated using the frequency of the source and the values of L and C.

Resonance in RLC Circuits

Resonance occurs in RLC circuits when the inductive reactance equals the capacitive reactance, resulting in maximum current flow and voltage being in phase. This condition is characterized by a specific frequency known as the resonant frequency. The formula for resonance involves the inductance and capacitance values, allowing for the calculation of the required capacitance to achieve resonance at a given frequency.