A current through a wire depends on time as
i = α₀t + βt²
where α₀ = 20 A/s and β = 8 As⁻². Find the charge crossed through a section of the wire in 15 s.

Solution

Physics sir · 3 steps

The current keeps changing, so do not multiply i by t. Integrate i over the 15 seconds.

1. Charge from current. Current is the rate of flow of charge, i = dq/dt. In a small time dt a charge dq = i dt crosses the section. Add up these pieces from t = 0 to t = 15 s.

   q = ∫₀¹⁵ (α₀ t + β t²) dt = [(α₀ t²)/2 + (β t³)/3]₀¹⁵

2. Put in numbers. First term: (20 × 225)/2 = 2250 C. Second term: (8 × 3375)/3 = 9000 C.
3. So. q = 2250 + 9000 = 11250 C, which is option (d).

The idea to keep. When current depends on time, charge is the area under the i–t graph: q = ∫ i dt. Use q = it only when the current is steady.

Answer from the NTA final answer key. Solution by PrepWiser.