The current through a wire depernds on time as `i=i_(0)+alphasin pit`, where `i_(0)=10A" and "alpha=pi/2`A. Find the charge crossed through a section of the wire in 3 seconds, and average current for that interval.

Answer

To solve the problem, we need to find the total charge that has crossed through a section of the wire in 3 seconds and the average current during that time interval.

### Step 1: Write down the expression for current
The current through the wire is given by:
\[ i(t) = i_0 + \alpha \sin(\pi t) \]
where \( i_0 = 10 \, \text{A} \) and \( \alpha = \frac{\pi}{2} \, \text{A} \).

### Step 2: Substitute the values of \( i_0 \) and \( \alpha \)
Substituting the given values into the equation:
\[ i(t) = 10 + \frac{\pi}{2} \sin(\pi t) \]

### Step 3: Find the expression for charge
The charge \( dq \) that flows through the wire in a small time interval \( dt \) is given by:
\[ dq = i(t) \, dt \]
To find the total charge \( Q \) that flows through the wire from \( t = 0 \) to \( t = 3 \) seconds, we integrate:
\[ Q = \int_0^3 i(t) \, dt = \int_0^3 \left( 10 + \frac{\pi}{2} \sin(\pi t) \right) dt \]

### Step 4: Perform the integration
We can split the integral:
\[ Q = \int_0^3 10 \, dt + \int_0^3 \frac{\pi}{2} \sin(\pi t) \, dt \]

Calculating the first integral:
\[ \int_0^3 10 \, dt = 10t \bigg|_0^3 = 10 \times 3 - 10 \times 0 = 30 \]

Now, calculating the second integral:
\[ \int_0^3 \frac{\pi}{2} \sin(\pi t) \, dt = \frac{\pi}{2} \left( -\frac{1}{\pi} \cos(\pi t) \right) \bigg|_0^3 = -\frac{1}{2} \left( \cos(3\pi) - \cos(0) \right) \]

Since \( \cos(3\pi) = -1 \) and \( \cos(0) = 1 \):
\[ = -\frac{1}{2} \left( -1 - 1 \right) = -\frac{1}{2} \times -2 = 1 \]

### Step 5: Combine the results
Now, combine both parts to find the total charge:
\[ Q = 30 + 1 = 31 \, \text{C} \]

### Step 6: Calculate the average current
The average current \( I_{avg} \) over the time interval can be calculated using:
\[ I_{avg} = \frac{Q}{\Delta t} = \frac{31 \, \text{C}}{3 \, \text{s}} = \frac{31}{3} \, \text{A} \approx 10.33 \, \text{A} \]

### Final Answer
The total charge that crossed through a section of the wire in 3 seconds is \( 31 \, \text{C} \) and the average current for that interval is \( \frac{31}{3} \, \text{A} \approx 10.33 \, \text{A} \).

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