A current in a wire is given by the equation, . `I=2t^(2)-3t+1`, the charge through cross section of : wire in time interval t=3s to t=5s is

Options

A. 32.33C
B. 43.34C
C. 45.5C
D. 42C

Explanation

To find the charge that passes through a cross-section of the wire from time \( t = 3 \) seconds to \( t = 5 \) seconds, we start with the given current equation:

\[ I = 2t^2 - 3t + 1 \]

The relationship between current \( I \) and charge \( Q \) is given by:

\[ I = \frac{dQ}{dt} \]

To find the charge \( Q \) that flows through the wire during the specified time interval, we need to integrate the current with respect to time:

\[ Q = \int_{t_1}^{t_2} I \, dt \]

where \( t_1 = 3 \) seconds and \( t_2 = 5 \) seconds. Thus, we can express the charge as:

\[ Q = \int_{3}^{5} (2t^2 - 3t + 1) \, dt \]

Now, we will compute the integral:

1. **Integrate the function**:
\[ \int (2t^2 - 3t + 1) \, dt = \frac{2}{3}t^3 - \frac{3}{2}t^2 + t + C \]

2. **Evaluate the definite integral from 3 to 5**:
\[ Q = \left[ \frac{2}{3}t^3 - \frac{3}{2}t^2 + t \right]_{3}^{5} \]

3. **Calculate at the upper limit \( t = 5 \)**:
\[ Q(5) = \frac{2}{3}(5^3) - \frac{3}{2}(5^2) + 5 \]
\[ = \frac{2}{3}(125) - \frac{3}{2}(25) + 5 \]
\[ = \frac{250}{3} - \frac{75}{2} + 5 \]
\[ = \frac{250}{3} - \frac{225}{6} + \frac{30}{6} \]
\[ = \frac{250}{3} - \frac{195}{6} \]
\[ = \frac{500}{6} - \frac{195}{6} = \frac{305}{6} \]

4. **Calculate at the lower limit \( t = 3 \)**:
\[ Q(3) = \frac{2}{3}(3^3) - \frac{3}{2}(3^2) + 3 \]
\[ = \frac{2}{3}(27) - \frac{3}{2}(9) + 3 \]
\[ = \frac{54}{3} - \frac{27}{2} + 3 \]
\[ = 18 - \frac{27}{2} + 3 \]
\[ = 21 - \frac{27}{2} = \frac{42}{2} - \frac{27}{2} = \frac{15}{2} \]

5. **Now, subtract the two results**:
\[ Q = Q(5) - Q(3) = \frac{305}{6} - \frac{15}{2} \]
\[ = \frac{305}{6} - \frac{45}{6} = \frac{260}{6} = \frac{130}{3} \]

Thus, the total charge that passes through the cross-section of the wire from \( t = 3 \) seconds to \( t = 5 \) seconds is:

\[ Q = \frac{130}{3} \text{ Coulombs} \approx 43.33 \text{ Coulombs} \]