Two neighbouring coils A and B have a mutual inductance of 20 mH. The current flowing through A is given by, `i = 3t^(2) - 4t + 6`. The induced eml at t=2s is

Options

A. 160 mV
B. 200 mV
C. 260 mV
D. 300 mV

Answer

To solve the problem, we need to find the induced electromotive force (emf) in coil B due to the changing current in coil A. The mutual inductance between the coils is given, and the current in coil A is expressed as a function of time. We will follow these steps:

### Step 1: Identify the given values
- Mutual inductance (M) = 20 mH = 20 × 10^(-3) H
- Current in coil A (i) = 3t² - 4t + 6

### Step 2: Differentiate the current with respect to time
To find the induced emf, we need to calculate the rate of change of current (di/dt).

1. Differentiate the current function:
\[ i(t) = 3t^2 - 4t + 6 \]
\[ \frac{di}{dt} = \frac{d}{dt}(3t^2 - 4t + 6) = 6t - 4 \]

### Step 3: Evaluate the derivative at t = 2 seconds
Now, we will substitute t = 2 seconds into the derivative to find the rate of change of current at that time.

1. Substitute t = 2 into di/dt:
\[ \frac{di}{dt} \bigg|_{t=2} = 6(2) - 4 = 12 - 4 = 8 \, \text{A/s} \]

### Step 4: Calculate the induced emf using the mutual inductance
The induced emf (ε) in coil B due to the changing current in coil A is given by the formula:
\[ \epsilon = -M \frac{di}{dt} \]
Substituting the values we have:
\[ \epsilon = - (20 \times 10^{-3}) \times (8) = -160 \times 10^{-3} \, \text{V} \]

### Step 5: Convert to millivolts
To express the induced emf in millivolts:
\[ \epsilon = -160 \, \text{mV} \]

### Final Answer
The induced emf at t = 2 seconds is **-160 mV**.

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