Number of electrons as a function of time

Question

Find the number of electrons as a function of time if:

i(t) = 10 cos(t), to and q(to) = 0

Answer 1

Integrate the current to find charge as a function of time, then divide the charge by e (the magnitude of the electron charge) to find the number of electrons as a function of time.

Assuming i(t) is in units of amps, the number of electrons is

N(t) = (1/e) q(t) = (1/e) ∫i(t)dt = (1/e) ∫10costdt = (1/e)(10sint + c)

q(t₀) = 0 ⇒ 10sint₀ + c = 0 ⇒ c = -10sint₀

N(t) = (1/e) (10sint - 10sint₀)