The magnetic flux across a loop of resistance `10Omega` is given by `phi=5t^(2)-4t+1Wb`. How much current is induced in the loop after `0.2` s?

Options

A. 0.4 A
B. 0.2 A
C. 0.04 A
D. 0.02 A

Explanation

To find the induced current in the loop after 0.2 seconds, we can follow these steps:

### Step 1: Write down the expression for magnetic flux.
The magnetic flux (φ) across the loop is given by:
\[ \phi = 5t^2 - 4t + 1 \, \text{Wb} \]

### Step 2: Differentiate the magnetic flux to find the induced EMF.
According to Faraday's law of electromagnetic induction, the induced electromotive force (EMF) (ε) is given by the negative rate of change of magnetic flux:
\[ \epsilon = -\frac{d\phi}{dt} \]
Now, we differentiate φ with respect to time (t):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(5t^2 - 4t + 1) = 10t - 4 \]
Thus, the induced EMF becomes:
\[ \epsilon = - (10t - 4) = -10t + 4 \]

### Step 3: Substitute the time value into the induced EMF equation.
We need to find the induced EMF at \( t = 0.2 \) seconds:
\[ \epsilon = -10(0.2) + 4 = -2 + 4 = 2 \, \text{V} \]

### Step 4: Calculate the induced current using Ohm's law.
Ohm's law states that the current (I) can be calculated using the formula:
\[ I = \frac{\epsilon}{R} \]
where R is the resistance of the loop. Given that the resistance \( R = 10 \, \Omega \):
\[ I = \frac{2 \, \text{V}}{10 \, \Omega} = 0.2 \, \text{A} \]

### Conclusion:
The induced current in the loop after 0.2 seconds is:
\[ I = 0.2 \, \text{A} \]

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