A series RLC circuit resonates at $7500 \ rad/s$ f

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A series RLC circuit resonates at $7500 \ rad/s$ for inductance $L = 20 \ mH$ and resistance $R = 10 \ \Omega$. The uncertainties in the measurement of $L$ and $R$ are $0.8 \ mH$ and $0.3 \ \Omega$, respectively. The percentage uncertainty in the measurement of quality factor is _________ $\%$ (rounded off to one decimal place).

The resonant frequency of an RLC circuit is given by the formula $\omega_0=\frac{1}{\sqrt{LC}}$. At resonance, the quality factor $Q$ is defined as $Q=\frac{\omega_0L}{R}$. Given $\omega_0=7500\ rad/s$, $L=20\ mH=0.02\ H$, and $R=10\ \Omega$, we first need to calculate $Q$.

The quality factor is:

$Q=\frac{7500 \times 0.02}{10}=15$

The uncertainty in $L$ is $\Delta L=0.8\ mH=0.0008\ H$, and the uncertainty in $R$ is $\Delta R=0.3\ \Omega$. Percentage uncertainties are calculated using:

For $L$: $\frac{\Delta L}{L} \times 100=\frac{0.0008}{0.02} \times 100=4\%$

For $R$: $\frac{\Delta R}{R} \times 100=\frac{0.3}{10} \times 100=3\%$

The percentage uncertainty in $Q$ is the quadratic sum of the individual percentage uncertainties:

$\sqrt{(4\%)^2 + (3\%)^2}=\sqrt{16 + 9}=\sqrt{25}=5\%$

The computed percentage uncertainty is $5\%$. This value is within the given range of 4.5 to 5.5. Thus, the percentage uncertainty in the measurement of quality factor is 5.0\%.

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