What is the total charge?

Question

The current in a circuit element varies according to the law I(t) = 1/sqrt(t) where the current I is measured in A, and time t is measured in sec. Find the total charge that has entered the element time during the period of time t with 1 ≤ t ≤ 4.

Answer 1

The current is the time rate of change of the electric charge - I = dQ/dt. We can write the charge differential dQ as I(t) dt such that integrating over the charge differential will yield the total charge Q = ∫ dQ = ∫ I(t) dt. Integrating from t = 1 to t = 4:

Q = ∫₁⁴ I(t) dt

Q = ∫₁⁴ t^-1/2 dt

Q = 2t^1/2 |₁⁴

Q = 2(4)^1/2 - 2(1)^1/2

Q = 2(2) - 2(1)

Q = 2 C

Answer 2

dq = I(t)dt = dt/√t; q = ∫₁⁴_dt/√t = 2√t₁⁴ = 4 - 2 = 2 C