The charge that has passed through a conductor since t =
The charge that has passed through a conductor since $t = 0$ is $Q = 8t^{2}-5t+1$ coulombs. Find the current at the end of the fifth second.
Explanation of Method
Current is the derivative of charge.
Step 1: The definition
Current is the rate at which charge passes a cross-section: $I(t) = \frac{dQ}{dt}$.
Step 2: Differentiate
\[ I(t) = \frac{d}{dt}\bigl(8t^{2}-5t+1\bigr) = 16t-5. \]
Step 3: Evaluate at $t = 5$
\[ I(5) = 80-5 = 75\ \text{A}. \]
Step 4: Verification
\[Q(5) = 200-25+1 = 176\ \text{C}, \qquad Q(4) = 128-20+1 = 109\ \text{C}\]
\[\text{Average current over the fifth second: } 176-109 = 67\ \text{A}, \text{ and } I(4.5) = 72-5 = 67\ \text{A} \;\;\checkmark\]
\[I = 0 \text{ at } t = \frac{5}{16} = 0.3125\ \text{s}, \text{ and } I\lt 0 \text{ before that} \;\;\checkmark\]
Result
Final answer:
\[\boxed{\,I(5) = 75\ \text{A}\,}\]
Remark
Remark. The current grows linearly, at $16\ \text{A}$ per second. Two features of $Q$ are worth noting as artefacts of the model rather than physics: $Q(0) = 1\ \text{C}$ is a reference offset, and the current is negative for $t\lt 0.3125\ \text{s}$, meaning the formula describes a real process only from that instant onward.