Question

The current in some DC circuits decays according to the function I=I0e−t/τ, where I is the current at some point in time, I0 is the initial current at time t=0, and τ is the time constant determined by the properties of the circuit. If you start such a circuit with an initial current of 1.2 Amps, and find that after 3.5 seconds have passed it has a current of 0.2 Amps, what is the value of the time constant τ for this particular circuit?

Answer 1

Answer: 1.95

Step-by-step explanation:

You should start off from the decay formula and solve for τ:

`I = I_(0)e^{(t)/(\tau\\  ) }`

`(I)/(I_(0)) = e^{(-t)/(\tau) }`

Apply inverse logarithmic function:

`ln((0.2 A)/(1.2 A) ) = (-t)/(\tau)`

The final form will be:

`\tau=(-3.5s)/(ln((0.2A)/(1.2A) ))`

Inputing values for I, IO, and t:

`\tau=(-3.5S)/(ln((0.2 A)/(1.2 A) )) = 1.95`