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

Verified step by step guidance

1. Calculate the natural angular frequency (${\omega }_0$) for the L-R-C circuit using the formula ${\omega }_0=\frac{1}{\sqrt{LC}}$, where L is the inductance and C is the capacitance.
2. Determine the decreased angular frequency (${\omega }^{\prime }$) which is 5% less than the natural angular frequency. Use the relation ${\omega }^{\prime }=0.95\times {\omega }_0$.
3. Use the formula for the angular frequency of a damped L-R-C circuit, $\omega =\sqrt{{\omega }_0^2-{\left(\frac{R}{2L}\right)}^2}$, and set $\omega ={\omega }^{\prime }$ to find the relationship involving R.
4. Solve the equation ${\omega }^{\prime }=\sqrt{{\omega }_0^2-{\left(\frac{R}{2L}\right)}^2}$ for R. Rearrange to get $R=2L\sqrt{{\omega }_0^2-{\omega }^{\prime 2}}$.
5. Substitute the values of L, ${\omega }_0$, and ${\omega }^{\prime }$ into the equation to find the required value of R.

Key Concepts

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an oscillating system, such as an L-R-C circuit, oscillates in radians per second. It is calculated using the formula ω = 1/√(LC) for a series circuit without resistance. In the presence of resistance, the angular frequency is affected, leading to a decrease in its value, which is crucial for understanding how R influences the circuit's behavior.

Damping in R-L-C Circuits

Damping refers to the effect of resistance in an R-L-C circuit, which reduces the amplitude of oscillations over time. The presence of resistance R introduces energy loss, which alters the natural frequency of the circuit. A specific value of R can be calculated to achieve a desired damping ratio, such as a 5.0% decrease in angular frequency, which is essential for controlling the circuit's response.

Resonance and Quality Factor

Resonance in an R-L-C circuit occurs when the inductive and capacitive reactances are equal, leading to maximum current flow at a specific frequency. The quality factor (Q) quantifies the sharpness of the resonance peak and is influenced by resistance. A decrease in angular frequency due to resistance affects the circuit's resonance characteristics, making it important to understand how R impacts overall circuit performance.