Question

A current through a wire depends on time as $i = \alpha_0 t + \beta t^2$ where $\alpha_0 = 20 \text{ A/s}$ and $\beta = 8 \text{ As}^{-2}$. Find the charge crossed through a section of the wire in 15 s.

• A. 260 C
• B. 2100 C
• C. 11250 C
• D. 2250 C

Hint:

Charge is the integral of current with respect to time. Graphically, it is the area under the $i-t$ curve.

Answer 1

The Correct Option is C

Step 1: $i = \frac{dq}{dt} \Rightarrow dq = i dt$.
Step 2: $q = \int_{0}^{15} (20t + 8t^2) dt$.
Step 3: $q = [10t^2 + \frac{8}{3}t^3]_{0}^{15}$.
Step 4: $q = 10(15^2) + \frac{8}{3}(15^3) = 10(225) + 8(15^2 \times 5) = 2250 + 8(1125)$.
Step 5: $q = 2250 + 9000 = 11250$ C.