The instantaneous flux associated with a closed circuit of `10 Omega` resistance is indicated by the following eqation `phi=6t^(2)+1`, then the value in amperes of the induced current at t = 0.25 sec will be:

Options

A. `1.2`
B. `0.8`
C. 6
D. `0.3`

Answer

To solve the problem, we need to find the induced current in a closed circuit with a given resistance and a time-dependent magnetic flux. The steps to solve the problem are as follows:

### Step 1: Understand the given data
- The resistance \( R = 10 \, \Omega \)
- The magnetic flux \( \phi(t) = 6t^2 + 1 \)

### Step 2: Calculate the induced electromotive force (emf)
The induced emf (\( E \)) in a circuit is given by Faraday's law of electromagnetic induction, which states that the induced emf is equal to the negative rate of change of magnetic flux through the circuit:

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

### Step 3: Differentiate the flux function
We need to differentiate the flux function \( \phi(t) = 6t^2 + 1 \):

\[ \frac{d\phi}{dt} = \frac{d}{dt}(6t^2 + 1) = 12t \]

Thus, the induced emf is:

\[ E = -12t \]

### Step 4: Calculate the induced current
The induced current \( I \) can be calculated using Ohm's law, which states that current is equal to the induced emf divided by the resistance:

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

Substituting the expression for \( E \):

\[ I = \frac{-12t}{10} \]

### Step 5: Substitute the value of \( t \)
Now, we will substitute \( t = 0.25 \, \text{sec} \) into the equation for current:

\[ I = \frac{-12(0.25)}{10} \]

Calculating this gives:

\[ I = \frac{-3}{10} = -0.3 \, \text{A} \]

Since current is typically expressed as a positive value (magnitude), we take the absolute value:

\[ I = 0.3 \, \text{A} \]

### Final Answer
The value of the induced current at \( t = 0.25 \, \text{sec} \) is \( 0.3 \, \text{A} \).

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