The current in a wire varies with time according to the relationship; \( I = (2 + 2t)\,\mathrm{A} \), where t is time in seconds. The charge passing through a cross-section of the wire in the time interval between \( t = 0\,\mathrm{s} \) to \( t = 4\,\mathrm{s} \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 24\,\mathrm{C} \)
Current is the rate of flow of charge, \( I = dq/dt \), so the charge that passes in a time interval is the integral of the current.
\( \displaystyle q = \int_{0}^{4}(2 + 2t)\,dt = \Big[\,2t + t^{2}\,\Big]_{0}^{4} = (8 + 16) - 0 = 24\,\mathrm{C} \)
Shortcut: The current is a straight line in time, so the charge is the average current times the interval. At \( t = 0 \) the current is 2 A and at \( t = 4\,\mathrm{s} \) it is 10 A, so the average is 6 A and \( q = 6 \times 4 = 24\,\mathrm{C} \).