Two colis, `1` and `2` have a mutual inductance `M = 5H` and resistance `R = 10 Omega` each. A current flows in coil 1, which varies with time as: `I_(1) = 2t^(2)`, where `t` is time. Find the total charge (in `C`) that has flows through coil `2` between `t = 0` and `t = 2 s`.

Explanation

To solve the problem step by step, we will follow the given information regarding the coils and the current flowing through coil 1.

### Step 1: Understand the given parameters
- Mutual inductance \( M = 5 \, \text{H} \)
- Resistance \( R = 10 \, \Omega \)
- Current in coil 1: \( I_1 = 2t^2 \)

### Step 2: Calculate the induced electromotive force (emf) in coil 2
The induced emf \( E \) in coil 2 due to the changing current in coil 1 can be calculated using the formula:
\[ E = -M \frac{dI_1}{dt} \]
First, we need to find \( \frac{dI_1}{dt} \):
\[ I_1 = 2t^2 \implies \frac{dI_1}{dt} = 4t \]
Now substituting this into the emf equation:
\[ E = -M \cdot 4t = -5 \cdot 4t = -20t \, \text{V} \]
(Note: The negative sign indicates the direction of induced emf, but we will consider the magnitude for further calculations.)

### Step 3: Calculate the induced current \( I \) in coil 2
Using Ohm's law, the induced current \( I \) in coil 2 can be calculated as:
\[ I = \frac{E}{R} = \frac{20t}{10} = 2t \, \text{A} \]

### Step 4: Calculate the total charge \( Q \) that flows through coil 2
The total charge \( Q \) that flows through coil 2 can be found by integrating the current \( I \) over the time interval from \( t = 0 \) to \( t = 2 \):
\[ Q = \int_{0}^{2} I \, dt = \int_{0}^{2} 2t \, dt \]
Calculating the integral:
\[ Q = 2 \int_{0}^{2} t \, dt = 2 \left[ \frac{t^2}{2} \right]_{0}^{2} = 2 \left[ \frac{2^2}{2} - \frac{0^2}{2} \right] = 2 \left[ \frac{4}{2} \right] = 2 \cdot 2 = 4 \, \text{C} \]

### Final Answer
The total charge that has flowed through coil 2 between \( t = 0 \) and \( t = 2 \, \text{s} \) is \( Q = 4 \, \text{C} \).

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