The electric current through a wire varies with time as I=I₀+β t, where I₀=20 A and β=3 A / s. The amount of electric charge crossed through a section of the wire in 20 s is :

Solution

Physics sir · 4 steps

The current keeps changing, so you cannot just do I times t: add up I dt over the 20 seconds.

1. Charge from current. Current is the rate of flow of charge, I = dq/dt. In a short time dt the charge that crosses is dq = I dt. Since I changes with time, add these small pieces by integrating.

   q = ∫₀²⁰ (I₀ + β t) dt = I₀ t + 1/2β t² |₀²⁰

2. Put in numbers. I₀ t = 20 × 20 = 400 C and 1/2β t² = 1/2 × 3 × 400 = 600 C.
3. Quick check. The current rises linearly from 20 A to 80 A, so its average is 50 A. Then 50 × 20 = 1000 C, the same result.
4. So. q = 400 + 600 = 1000 C, which is option (c).

The idea to keep. Charge is the area under the I–t graph. When the current changes with time, integrate it, or for a linear change multiply the average current by the time.

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