An L-R-C series circuit has L = 0.450 H, C = 2.50 × 10^-5 F, and resistance R. What value must R have to give a 5.0% decrease in angular frequency compared to the value calculated in part (a)?

검증된 단계별 안내

First, recall the formula for the angular frequency \( \omega_0 \) of an L-R-C series circuit without resistance: \( \omega_0 = \frac{1}{\sqrt{LC}} \). Substitute the given values for L and C to find \( \omega_0 \).

Next, understand that the presence of resistance R in the circuit modifies the angular frequency to \( \omega = \sqrt{\omega_0^2 - \left(\frac{R}{2L}\right)^2} \). This is the formula for the damped angular frequency.

To achieve a 5.0% decrease in angular frequency, set \( \omega = 0.95 \omega_0 \). Substitute \( \omega_0 \) from step 1 into this equation.

Rearrange the equation \( \omega = \sqrt{\omega_0^2 - \left(\frac{R}{2L}\right)^2} = 0.95 \omega_0 \) to solve for R. This involves isolating \( \left(\frac{R}{2L}\right)^2 \) and then taking the square root.

Finally, substitute the known values for L and \( \omega_0 \) into the rearranged equation to find the value of R that results in the desired decrease in angular frequency.

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주요 개념

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

Angular Frequency in L-R-C Circuits

Angular frequency in an L-R-C circuit is determined by the formula ω = 1/√(LC), where L is inductance and C is capacitance. This frequency represents the rate of oscillation of the circuit's current and voltage. In the context of the question, understanding how resistance affects this frequency is crucial for determining the required decrease.

Damping in RLC Circuits

Damping refers to the reduction in amplitude of oscillations in a circuit due to resistance. In an L-R-C circuit, resistance causes energy loss, affecting the angular frequency. The damping factor, which includes resistance, modifies the natural frequency, leading to a decrease in angular frequency as resistance increases.

Quality Factor and Resonance

The quality factor (Q) of an L-R-C circuit is a measure of its resonance sharpness, defined as Q = ωL/R. It indicates how underdamped the circuit is, affecting the angular frequency. A decrease in Q due to increased resistance results in a lower angular frequency, which is essential for calculating the required resistance for a specific frequency change.