The current through a wire depends on time as, `i=(10+4t)` Here , `i` is ampere and `t` in seconds. Find the charge crossed through section in time interval between `t=0` to `t=10s`.

Explanation

To find the charge that has crossed through a section of wire in the time interval from \( t = 0 \) to \( t = 10 \) seconds, we will use the relationship between current and charge. The current \( i(t) \) is given as:

\[ i(t) = 10 + 4t \quad \text{(in amperes)} \]

The charge \( Q \) that flows through a section of the wire can be calculated using the integral of the current over time:

\[ Q = \int_{t_1}^{t_2} i(t) \, dt \]

### Step 1: Set up the integral
We need to evaluate the integral from \( t = 0 \) to \( t = 10 \):

\[ Q = \int_{0}^{10} (10 + 4t) \, dt \]

### Step 2: Split the integral
We can split the integral into two parts:

\[ Q = \int_{0}^{10} 10 \, dt + \int_{0}^{10} 4t \, dt \]

### Step 3: Evaluate the first integral
The first integral is straightforward:

\[ \int_{0}^{10} 10 \, dt = 10 \cdot t \bigg|_{0}^{10} = 10 \cdot 10 - 10 \cdot 0 = 100 \]

### Step 4: Evaluate the second integral
Now, we evaluate the second integral:

\[ \int_{0}^{10} 4t \, dt = 4 \cdot \frac{t^2}{2} \bigg|_{0}^{10} = 2t^2 \bigg|_{0}^{10} = 2 \cdot (10^2) - 2 \cdot (0^2) = 2 \cdot 100 = 200 \]

### Step 5: Combine the results
Now, we combine the results from both integrals:

\[ Q = 100 + 200 = 300 \, \text{coulombs} \]

### Final Answer
The total charge that has crossed through the section in the time interval from \( t = 0 \) to \( t = 10 \) seconds is:

\[ \boxed{300 \, \text{coulombs}} \]

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