It is known to all of you that the impedance of a circuit is dependent on the frequency of source. In order to study the effect of frequency on the impedance, a student in a lab took 2 impedance boxes `P` and `Q` and connected them in series with an `AC` source of variable frequency. The emf of the source is constant at `10 V` Box `P` contains a capacitance of `1muF` in series with a resistance of `32 Omega`. And the box `Q` has a coil of self-inductance `4.9 mH` and a resistance of `68 Omega`in series. He adjusted the frequency so that the maximum current flows in `P` and `Q`. Based on his experimental set up and the reading by him at various moment, answer the following questions.
Power factor of the circuit at maximum current is

Options

A. `1/2`
B. 1
C. 0
D. `1/sqrt(2)`

Explanation

To find the power factor of the circuit at maximum current, we can follow these steps:

### Step 1: Understand the Condition for Maximum Current
At maximum current, the circuit is at resonance. This means that the inductive reactance (XL) and capacitive reactance (XC) are equal.

### Step 2: Identify the Components in the Circuit
- For Box P:
- Capacitance (C) = 1 µF = 1 × 10^-6 F
- Resistance (R1) = 32 Ω

- For Box Q:
- Inductance (L) = 4.9 mH = 4.9 × 10^-3 H
- Resistance (R2) = 68 Ω

### Step 3: Calculate the Reactances
- The capacitive reactance (XC) can be calculated using the formula:
\[ X_C = \frac{1}{2 \pi f C} \]

- The inductive reactance (XL) can be calculated using the formula:
\[ X_L = 2 \pi f L \]

### Step 4: Set the Reactances Equal at Resonance
At resonance, we have:
\[ X_L = X_C \]
This means:
\[ 2 \pi f L = \frac{1}{2 \pi f C} \]

### Step 5: Calculate the Total Resistance
The total resistance (R_total) in the circuit is the sum of the resistances from both boxes:
\[ R_{total} = R_1 + R_2 = 32 \, \Omega + 68 \, \Omega = 100 \, \Omega \]

### Step 6: Calculate the Total Impedance at Resonance
At resonance, the total impedance (Z) is equal to the total resistance:
\[ Z = R_{total} = 100 \, \Omega \]

### Step 7: Calculate the Power Factor
The power factor (PF) is given by:
\[ PF = \cos \phi = \frac{R}{Z} \]
Substituting the values:
\[ PF = \frac{100 \, \Omega}{100 \, \Omega} = 1 \]

### Conclusion
The power factor of the circuit at maximum current is **1**.

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