To solve the problem step by step, we will analyze the given magnetic flux equation and calculate the induced current, charge flow, and check the validity of the options provided.
### Given:
- Magnetic flux, \( \phi(t) = 2t^2 - 4 \) Weber
- Resistance of the coil, \( R = 10 \, \Omega \)
### Step 1: Calculate the induced EMF
The induced EMF (\( \mathcal{E} \)) can be calculated using Faraday's law of electromagnetic induction, which states that the induced EMF is equal to the rate of change of magnetic flux:
\[
\mathcal{E} = -\frac{d\phi}{dt}
\]
First, we need to differentiate the magnetic flux with respect to time \( t \):
\[
\frac{d\phi}{dt} = \frac{d}{dt}(2t^2 - 4) = 4t
\]
Thus, the induced EMF is:
\[
\mathcal{E} = -4t
\]
### Step 2: Calculate the induced current at \( t = 2 \) seconds
Now, we will find the induced current (\( I \)) using Ohm's law:
\[
I = \frac{\mathcal{E}}{R}
\]
Substituting \( t = 2 \) seconds into the EMF equation:
\[
\mathcal{E} = -4(2) = -8 \, \text{V}
\]
Now, using the absolute value of EMF (since current is a scalar quantity):
\[
I = \frac{8}{10} = 0.8 \, \text{A}
\]
### Step 3: Check if the induced current increases linearly with time
From the EMF equation, we have:
\[
\mathcal{E} = 4t
\]
Thus, the induced current can be expressed as:
\[
I = \frac{4t}{10} = 0.4t
\]
This shows that the induced current is directly proportional to time \( t \), indicating that it increases linearly with time.
### Step 4: Calculate the charge flow through the coil from \( t = 0 \) to \( t = 2 \) seconds
The charge (\( Q \)) flowing through the coil can be calculated using the formula:
\[
Q = \int_0^t I \, dt
\]
Using the expression for current:
\[
Q = \int_0^2 0.4t \, dt
\]
Calculating the integral:
\[
Q = 0.4 \int_0^2 t \, dt = 0.4 \left[ \frac{t^2}{2} \right]_0^2 = 0.4 \left[ \frac{2^2}{2} - 0 \right] = 0.4 \left[ 2 \right] = 0.8 \, \text{C}
\]
### Step 5: Conclusion on the options
- **Option A**: The induced current at \( t = 2 \) seconds is \( 0.8 \, \text{A} \) (Correct)
- **Option B**: The induced current increases linearly with time (Correct)
- **Option C**: The charge flow through the coil from \( t = 0 \) to \( t = 2 \) seconds is \( 0.8 \, \text{C} \) (Correct)
- **Option D**: The net flow of charge is zero (Incorrect, as the net charge is \( 0.8 \, \text{C} \))
### Final Answer:
Options A, B, and C are correct, while option D is incorrect.
---
