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

Explanation:

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

$I = I_{0}e^{\frac{t}{\tau\\ } }$

$\frac{I}{I_{0}} = e^{\frac{-t}{\tau} }$

Apply inverse logarithmic function:

$ln(\frac{0.2 A}{1.2 A} ) = \frac{-t}{\tau}$

The final form will be:

$\tau=\frac{-3.5s}{ln(\frac{0.2A}{1.2A} )}$

Inputing values for I, IO, and t:

$\tau=\frac{-3.5S}{ln(\frac{0.2 A}{1.2 A} )} = 1.95$