Question

A conductor carries a current that is decreasing exponentially with time. The current is modeled as  I = I₀e^−t/τ, where I₀ = 6.50 A is the current at time

t = 0.00 s

and

τ = 0.50 s

is the time constant. How much charge flows through the conductor (in C) between

t = 0.00 s

and

t = 4τ?

(Enter the magnitude.)

C

Please show all the work done

Answer 1

b = 0.5 s (one time constant)

a = 6.50 A (initial current)

Exponential decay of current is given by:

i(t) = a*e^-(t/b) = dQ/dt

dQ = [ a*e^-(t/b)]*dt

integrate from t = 0 to t=4 time constants:

Q = ∫ [a*e^(-t/b)] dt = -abe^(-t/b) |0,4b

Q = 3.19 C