Question

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 the quality factor is ________ % (rounded off to one decimal place).

Hint:

When calculating the uncertainty in the quality factor of a series RLC circuit, use the propagation of uncertainties formula, and remember to apply the formula for \( Q = \frac{\omega_0 L}{R} \).

Answer 1

Step 1: Formula for the quality factor \( Q \).
The quality factor \( Q \) for a series RLC circuit is given by the formula: \[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \] where \( L \) is the inductance, \( R \) is the resistance, and \( C \) is the capacitance. At resonance, the angular frequency \( \omega_0 \) is related to \( L \) and \( C \) by: \[ \omega_0 = \frac{1}{\sqrt{LC}} \] Therefore, \( C \) can be expressed as: \[ C = \frac{1}{L \omega_0^2} \] Step 2: Quality factor formula using \( \omega_0 \).
Substitute the expression for \( C \) into the equation for \( Q \): \[ Q = \frac{\omega_0 L}{R} \] Step 3: Apply the given values.
We are given:
\( L = 20 \, {mH} = 20 \times 10^{-3} \, {H} \),
\( R = 10 \, \Omega \),
\( \omega_0 = 7500 \, {rad/s} \).
Thus, the quality factor \( Q \) is: \[ Q = \frac{7500 \times 20 \times 10^{-3}}{10} = \frac{150}{10} = 15 \] Step 4: Calculate the uncertainty in \( Q \).
The uncertainty in the quality factor is found by using the propagation of uncertainties for the formula \( Q = \frac{\omega_0 L}{R} \): \[ \left( \frac{\Delta Q}{Q} \right)^2 = \left( \frac{\Delta L}{L} \right)^2 + \left( \frac{\Delta R}{R} \right)^2 \] where \( \Delta L = 0.8 \, {mH} = 0.8 \times 10^{-3} \, {H} \) and \( \Delta R = 0.3 \, \Omega \). Substitute the given values into the uncertainty formula: \[ \frac{\Delta Q}{Q} = \sqrt{\left( \frac{0.8 \times 10^{-3}}{20 \times 10^{-3}} \right)^2 + \left( \frac{0.3}{10} \right)^2} \] \[ \frac{\Delta Q}{Q} = \sqrt{\left( 0.04 \right)^2 + \left( 0.03 \right)^2} \] \[ \frac{\Delta Q}{Q} = \sqrt{0.0016 + 0.0009} = \sqrt{0.0025} = 0.05 \] Step 5: Calculate the percentage uncertainty.
The percentage uncertainty in \( Q \) is: \[ % \, \Delta Q = 0.05 \times 100 = 5% \] Step 6: Final Answer.
Thus, the percentage uncertainty in the measurement of the quality factor is: \[ \boxed{5%} \]