The electric current through a wire varies with time as `I=I_(0)+beta t` ,where `I_(0)=20A]` and `beta=3A/s` The amount of electric charge crossed through a section of the wire , in 20 s is:

Options

A. 1000 C
B. 80 C
C. 800 C
D. 1600 C

Answer

To find the amount of electric charge that has crossed through a section of the wire in 20 seconds, we start with the given current as a function of time:

\[ I(t) = I_0 + \beta t \]

where \( I_0 = 20 \, \text{A} \) and \( \beta = 3 \, \text{A/s} \).

### Step-by-step Solution:

1. **Write the expression for current**:
\[ I(t) = 20 + 3t \]

2. **Recall the relationship between current and charge**:
The electric charge \( Q \) that passes through a section of the wire over a time interval can be found by integrating the current over that time period:
\[ Q = \int_0^T I(t) \, dt \]
where \( T \) is the time duration (in this case, \( T = 20 \, \text{s} \)).

3. **Set up the integral**:
\[ Q = \int_0^{20} (20 + 3t) \, dt \]

4. **Integrate the function**:
We can split the integral:
\[ Q = \int_0^{20} 20 \, dt + \int_0^{20} 3t \, dt \]

- The first integral:
\[ \int_0^{20} 20 \, dt = 20t \bigg|_0^{20} = 20 \times 20 - 20 \times 0 = 400 \]

- The second integral:
\[ \int_0^{20} 3t \, dt = 3 \cdot \frac{t^2}{2} \bigg|_0^{20} = 3 \cdot \frac{20^2}{2} - 3 \cdot \frac{0^2}{2} = 3 \cdot \frac{400}{2} = 3 \cdot 200 = 600 \]

5. **Combine the results**:
\[ Q = 400 + 600 = 1000 \, \text{C} \]

Thus, the total electric charge that crossed through a section of the wire in 20 seconds is:

\[ \boxed{1000 \, \text{C}} \]