The mutual inductance between a primary and secondary circuit is `0.5H`. The resistance of the primary and the secondary circuits are `20 ohms` and `5ohms` respectvely. To genrate a current of `0.4A` in the secondary,current in the primary must be changed at the rate of

Options

A. `4.0 A//s`
B. `16.0 A//s`
C. `1.6 A//s`
D. `8.0 A//s`

Explanation

To solve the problem, we need to find the rate of change of current in the primary circuit (di/dt) that is required to generate a current of 0.4 A in the secondary circuit. We will use the concept of mutual inductance and Ohm's law.

### Step-by-Step Solution:

1. **Understand the Relationship**: The mutual inductance (M) relates the change in current in the primary circuit to the induced electromotive force (emf) in the secondary circuit. The formula for induced emf (ε) in the secondary circuit due to the change in current in the primary circuit is given by:
\[ \epsilon = M \frac{di}{dt} \]

2. **Apply Ohm's Law**: The induced emf in the secondary circuit can also be expressed using Ohm's law:
\[ \epsilon = I \cdot R \]
where \(I\) is the current in the secondary circuit and \(R\) is the resistance of the secondary circuit.

3. **Set Up the Equation**: Since both expressions represent the induced emf, we can set them equal to each other:
\[ M \frac{di}{dt} = I \cdot R \]

4. **Substitute Known Values**: We know the following values:
- Mutual inductance, \(M = 0.5 \, H\)
- Current in the secondary circuit, \(I = 0.4 \, A\)
- Resistance of the secondary circuit, \(R = 5 \, \Omega\)

Plugging these values into the equation gives:
\[ 0.5 \frac{di}{dt} = 0.4 \cdot 5 \]

5. **Calculate the Right Side**: Calculate the right side of the equation:
\[ 0.4 \cdot 5 = 2 \, V \]
So, we have:
\[ 0.5 \frac{di}{dt} = 2 \]

6. **Solve for di/dt**: Now, isolate \(\frac{di}{dt}\):
\[ \frac{di}{dt} = \frac{2}{0.5} = 4 \, A/s \]

### Final Answer:
The rate of change of current in the primary circuit must be \(4 \, A/s\).

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