A series LCR circuit has resistance 10Ω and reactance is 7√2 Ω. What is the impedance of the circuit?

Answer

To find the impedance of a series LCR circuit, we can use the formula for impedance \( Z \):

\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

Where:
- \( R \) is the resistance,
- \( X_L \) is the inductive reactance,
- \( X_C \) is the capacitive reactance.

In this problem, we are given:
- Resistance \( R = 10 \, \Omega \)
- Reactance \( |X| = 7\sqrt{2} \, \Omega \)

Since we are not given \( X_L \) and \( X_C \) separately, we can assume that the given reactance \( |X| \) represents the net reactance \( |X_L - X_C| \).

### Step 1: Calculate \( R^2 \)
\[ R^2 = (10)^2 = 100 \]

### Step 2: Calculate \( (X_L - X_C)^2 \)
\[ (X_L - X_C)^2 = (7\sqrt{2})^2 = 49 \times 2 = 98 \]

### Step 3: Substitute values into the impedance formula
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{100 + 98} \]

### Step 4: Simplify the expression
\[ Z = \sqrt{198} \]

### Step 5: Calculate the square root
\[ Z \approx 14.07 \, \Omega \]

Thus, the impedance of the circuit is approximately \( 14.07 \, \Omega \).

---