In a series LCR circuit resistance `R = 10Omega` and the impedance `Z = 10 Omega` The phase difference between the current and the voltage is

Options

A. `0^@`
B. `30^@`
C. `45^@`
D. `60^@`

Answer

To find the phase difference between the current and the voltage in a series LCR circuit, we can follow these steps:

### Step 1: Understand the relationship between resistance (R), impedance (Z), and phase difference (φ)
In a series LCR circuit, the impedance (Z) is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \(X_L\) is the inductive reactance and \(X_C\) is the capacitive reactance.

### Step 2: Given values
From the problem, we know:
- Resistance \(R = 10 \, \Omega\)
- Impedance \(Z = 10 \, \Omega\)

### Step 3: Substitute the values into the impedance formula
Since \(Z = 10 \, \Omega\) and \(R = 10 \, \Omega\), we can substitute these values into the impedance formula:
\[ 10 = \sqrt{10^2 + (X_L - X_C)^2} \]

### Step 4: Simplify the equation
Squaring both sides gives:
\[ 100 = 100 + (X_L - X_C)^2 \]
This simplifies to:
\[ 0 = (X_L - X_C)^2 \]

### Step 5: Solve for the reactance
Taking the square root of both sides, we find:
\[ X_L - X_C = 0 \]
This implies:
\[ X_L = X_C \]

### Step 6: Determine the phase difference
In a series LCR circuit, the phase difference (φ) between the current and voltage can be calculated using:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]
Since we found that \(X_L - X_C = 0\):
\[ \tan(\phi) = \frac{0}{10} = 0 \]

### Step 7: Calculate φ
The phase difference φ is:
\[ \phi = \tan^{-1}(0) = 0 \]

### Conclusion
The phase difference between the current and the voltage in the circuit is:
\[ \phi = 0 \, \text{radians} \]