### Correct Option: B. The quality factor decreases ### Detailed Explanation: In a series RLC circuit, the quality factor (Q) is a dimensionless parameter that describes how underdamped the circuit is and is defined as the ratio of the resonant frequency to the bandwidth of the circuit. The formula for the quality factor \( Q \) in a series RLC circuit is given by: \[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \] Where: - \( R \) is the resistance in ohms (Ω), - \( L \) is the inductance in henries (H), - \( C \) is the capacitance in farads (F). #### Step-by-Step Analysis: 1. **Understanding the Formula**: - The quality factor \( Q \) is inversely proportional to the resistance \( R \). This means that as resistance increases, the value of \( Q \) decreases. - The term \( \sqrt{\frac{L}{C}} \) is a constant for a given circuit, as \( L \) and \( C \) do not change when we vary \( R \). 2. **Effect of Increasing Resistance**: - When you increase the resistance \( R \), the denominator in the formula for \( Q \) increases. Since \( Q \) is inversely proportional to \( R \), this results in a decrease in the value of \( Q \). - Physically, a higher resistance means that the circuit dissipates more energy as heat, which leads to a broader bandwidth and thus a lower quality factor. 3. **Resonant Frequency and Bandwidth**: - The resonant frequency \( f_0 \) of the circuit is given by: \[ f_0 = \frac{1}{2\pi\sqrt{LC}} \] - The bandwidth \( \Delta f \) is related to the quality factor by: \[ \Delta f = \frac{f_0}{Q} \] - As \( R \) increases, \( Q \) decreases, which means \( \Delta f \) increases. This indicates that the circuit is less selective and has a wider range of frequencies over which it can operate effectively. ### Why Other Options Are Incorrect: - **Option A: The quality factor increases**: This is incorrect because increasing resistance leads to a decrease in \( Q \), as explained above. Higher resistance means more energy loss, which reduces the sharpness of the resonance peak. - **Option C: The quality factor remains unchanged**: This is also incorrect. The quality factor is directly affected by the resistance in the circuit. If resistance changes, \( Q \) changes as well. - **Option D: The quality factor doubles**: This is incorrect because there is no direct relationship that would cause \( Q \) to double with an increase in resistance. In fact, \( Q \) decreases as resistance increases. ### Common Pitfalls: - **Confusing Q with other parameters**: Students often confuse the quality factor with other circuit parameters. Remember that \( Q \) specifically relates to the energy loss in the circuit and its resonance characteristics. - **Ignoring the relationship between R, L, and C**: It’s crucial to remember that \( Q \) is dependent on the values of resistance, inductance, and capacitance. Changes in one can affect the overall behavior of the circuit. ### Revision Summary: - The quality factor \( Q \) in a series RLC circuit is inversely proportional to resistance \( R \). - Increasing resistance leads to a decrease in \( Q \), indicating more energy loss and a broader bandwidth. - The formula for \( Q \) is \( Q = \frac{1}{R} \sqrt{\frac{L}{C}} \). - Higher resistance results in a less selective circuit with a wider range of frequencies.