Question

Problem 1 (5 points). Find the number of electrons as a function of time if: ķ unly 10 Sint → 2pt i(t) = 10 cos(t), to and q(to) = 0

Answer 1

Our current varies with time according to the law

$$i(t)=10\text{cos}t.$$

We know that the current is

$$i(t)=\frac{dq(t)}{dt},\\ dq(t)=i(t)dt,\\ dq(t)=10\text{cos}t\text{d}t,\\ q(t)=\int10\text{cos}t\text{d}t=10\text{sin}t+C=10\text{sin}t$$

because $C=q(t_0) = 0.$

On the other hand, we know that this charge was produced by electrons charged $e$ coulombs each:

$$q(t)=N(t)e,\\ \space\\ N(t)=\frac{q(t)}{e}=\frac{10}{e}\text{sin}t.$$