The magnetic flux `(phi)` in a closed circuit of resistance `20 Omega` varies with time (t) according to the equation `phi = 7t^(2) - 4t` where `phi` is in weber and t is in seconds. The magnitude of the induced current at t =0.25s is

Options

A. 25mA
B. `0.025 mA`
C. 47 mA
D. 175 mA

Answer

To solve the problem, we need to find the induced current in the circuit at time \( t = 0.25 \) seconds. We will follow these steps:

### Step 1: Write down the given equation for magnetic flux
The magnetic flux \( \phi \) is given by the equation:
\[ \phi = 7t^2 - 4t \]

### Step 2: Differentiate the magnetic flux to find the induced emf
The induced emf \( \mathcal{E} \) is given by Faraday's law of electromagnetic induction, which states:
\[ \mathcal{E} = -\frac{d\phi}{dt} \]
Now, we differentiate \( \phi \) with respect to \( t \):
\[ \frac{d\phi}{dt} = \frac{d}{dt}(7t^2 - 4t) = 14t - 4 \]
Thus, the induced emf is:
\[ \mathcal{E} = - (14t - 4) = -14t + 4 \]

### Step 3: Substitute \( t = 0.25 \) seconds into the emf equation
Now, we will substitute \( t = 0.25 \) seconds into the equation for induced emf:
\[ \mathcal{E} = -14(0.25) + 4 \]
Calculating this gives:
\[ \mathcal{E} = -3.5 + 4 = 0.5 \text{ volts} \]

### Step 4: Use Ohm's law to find the induced current
Ohm's law states that:
\[ I = \frac{\mathcal{E}}{R} \]
where \( R \) is the resistance of the circuit. Given that \( R = 20 \, \Omega \):
\[ I = \frac{0.5}{20} \]
Calculating this gives:
\[ I = 0.025 \text{ amperes} \]

### Step 5: Convert the current to milliamperes
To convert amperes to milliamperes, we multiply by 1000:
\[ I = 0.025 \times 1000 = 25 \text{ milliamperes} \]

### Final Answer
The magnitude of the induced current at \( t = 0.25 \) seconds is:
\[ \boxed{25 \text{ mA}} \]