A coil of resistance `400Omega` is placed in a magnetic field. If the magnetic flux `phi` (wb) linked with the coil varies with time `t` (sec) as `f=50t^(2)+4`, the current in the coil at `t=2` sec is

Options

A. a. 2A
B. b. 1A
C. c. 0.5 A
D. d. 0.1 A

Explanation

To solve the problem step by step, we need to find the current in the coil at time \( t = 2 \) seconds given the magnetic flux \( \phi(t) = 50t^2 + 4 \) and the resistance \( R = 400 \, \Omega \).

### Step 1: Calculate the induced electromotive force (emf)
The induced emf \( \mathcal{E} \) in the coil can be calculated using Faraday's law of electromagnetic induction, which states:

\[ \mathcal{E} = -\frac{d\phi}{dt} \]

First, we need to differentiate the magnetic flux \( \phi(t) \) with respect to time \( t \).

Given:
\[ \phi(t) = 50t^2 + 4 \]

Now, differentiate \( \phi(t) \):

\[ \frac{d\phi}{dt} = \frac{d}{dt}(50t^2 + 4) = 100t \]

### Step 2: Substitute \( t = 2 \) seconds into the derivative
Now, we substitute \( t = 2 \) seconds into the derivative to find the induced emf:

\[ \mathcal{E} = -\frac{d\phi}{dt} = -100t \]

Substituting \( t = 2 \):

\[ \mathcal{E} = -100 \times 2 = -200 \, \text{V} \]

Since we are interested in the magnitude of the emf, we take:

\[ \mathcal{E} = 200 \, \text{V} \]

### Step 3: Use Ohm's Law to find the current
According to Ohm's law, the current \( I \) through the coil can be calculated using the formula:

\[ I = \frac{V}{R} \]

Where \( V \) is the induced emf and \( R \) is the resistance of the coil.

Given:
- \( V = 200 \, \text{V} \)
- \( R = 400 \, \Omega \)

Now, substitute these values into the formula:

\[ I = \frac{200}{400} = 0.5 \, \text{A} \]

### Final Answer
The current in the coil at \( t = 2 \) seconds is:

\[ I = 0.5 \, \text{A} \]

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