A coil has resistance `30 ohm` and inductive reactance `20 ohm` at `50 Hz` frequency. If an ac source of 200 volts. `100 Hz`, is connected across the coil, the current in the coil will be

Options

A. 2.0 A
B. 4.0 A
C. 8.0 A
D. `(20)/(sqrt13)` A

Explanation

To solve the problem step by step, we will use the given information about the coil's resistance and inductive reactance, and the AC source voltage and frequency.

### Step 1: Identify the given values
- Resistance (R) = 30 ohms
- Inductive reactance at 50 Hz (X_L1) = 20 ohms
- Frequency 1 (f1) = 50 Hz
- Frequency 2 (f2) = 100 Hz
- AC source voltage (V_rms) = 200 volts

### Step 2: Calculate the inductive reactance at the new frequency (100 Hz)
Inductive reactance (X_L) is directly proportional to frequency. We can use the ratio of the frequencies to find the new inductive reactance (X_L2) at 100 Hz.

Using the formula:
\[ \frac{X_{L1}}{X_{L2}} = \frac{f1}{f2} \]
Substituting the known values:
\[ \frac{20}{X_{L2}} = \frac{50}{100} \]
Cross-multiplying gives:
\[ 20 \cdot 100 = 50 \cdot X_{L2} \]
\[ 2000 = 50 \cdot X_{L2} \]
Now, solving for \(X_{L2}\):
\[ X_{L2} = \frac{2000}{50} = 40 \text{ ohms} \]

### Step 3: Calculate the impedance (Z) of the coil
The impedance (Z) of a coil can be calculated using the formula:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values we have:
\[ Z = \sqrt{30^2 + 40^2} \]
Calculating the squares:
\[ Z = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ ohms} \]

### Step 4: Calculate the current (I) in the coil
Using Ohm's law for AC circuits, the current can be calculated using:
\[ I = \frac{V_{rms}}{Z} \]
Substituting the known values:
\[ I = \frac{200}{50} = 4 \text{ amperes} \]

### Final Answer
The current in the coil will be **4 amperes**.

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