To solve the problem step by step, we will analyze the behavior of the R-L-C series circuit when the resistance is doubled.
### Step 1: Understand the Initial Conditions
In the given R-L-C series circuit, the potential difference (PD) across each element (R, L, and C) is initially 20 V. This indicates that the circuit is in resonance, where:
- \( V_R = V_L = V_C = 20 \, \text{V} \)
### Step 2: Analyze the Effect of Doubling the Resistance
When the resistance \( R \) is doubled, the new resistance becomes \( R' = 2R \).
### Step 3: Determine the Current in the Circuit
In a series circuit, the current \( I \) is given by:
\[ I = \frac{V}{Z} \]
Where \( Z \) is the impedance of the circuit. At resonance:
\[ Z = R \]
Thus, the initial current \( I \) can be expressed as:
\[ I = \frac{V}{R} \]
Given that \( V = 20 \, \text{V} \), we have:
\[ I = \frac{20}{R} \]
### Step 4: Calculate the New Current After Doubling Resistance
After doubling the resistance, the new current \( I' \) becomes:
\[ I' = \frac{V}{R'} = \frac{V}{2R} = \frac{20}{2R} = \frac{10}{R} \]
This shows that the new current is half of the original current:
\[ I' = \frac{I}{2} \]
### Step 5: Calculate the New Potential Differences
Using Ohm's Law, the potential difference across each component can be calculated:
1. **For the resistor (R)**:
\[ V_R' = I' \cdot R' = \frac{10}{R} \cdot 2R = 20 \, \text{V} \]
2. **For the inductor (L)**:
Since the circuit is still in resonance, the potential difference across the inductor will equal the potential difference across the capacitor:
\[ V_L' = V_C' = \frac{I'}{\omega L} \]
But since \( I' = \frac{10}{R} \) and at resonance \( V_L = V_C \):
\[ V_L' = V_C' = 10 \, \text{V} \]
### Conclusion
After doubling the resistance, the potential differences across the components are:
- \( V_R' = 20 \, \text{V} \)
- \( V_L' = 10 \, \text{V} \)
- \( V_C' = 10 \, \text{V} \)
### Final Answer
The potential differences across R, L, and C respectively are:
- \( V_R = 20 \, \text{V} \)
- \( V_L = 10 \, \text{V} \)
- \( V_C = 10 \, \text{V} \)
