AP Physics 1 Exam Question:

A circuit consists of a battery with an emf of 9V connected in series with a resistor R and an inductor L, as shown in the diagram below. The resistance R is 3Ω and the inductance L is 0.2H.

a) If the switch is closed at time t = 0, determine the time constant τ of the circuit. b) Calculate the initial rate at which the current in the circuit is changing. c) Calculate the maximum current that will flow in the circuit. d) Find the expression for the current i(t) in the circuit as a function of time t, for t > 0.

Explanation:

a) To determine the time constant τ of the circuit, we'll use the equation τ = L/R, where L is the inductance and R is the resistance. Substituting the given values, we have:

τ = 0.2H / 3Ω = 0.0667 s

b) The initial rate at which the current is changing is given by the equation i(t) = (emf/R)(1 - e^(-t/τ)). The rate of change of current at t = 0 can be found by taking the derivative of this equation with respect to t and evaluating it at t = 0. Differentiating, we get:

di(t)/dt = (emf/R)(e^(-t/τ)/τ)

Substituting t = 0 and the given values, we have:

di(t)/dt | t=0 = (9V / 3Ω)(e^0 / 0.0667s) = 135 A/s

Therefore, the initial rate at which the current in the circuit is changing is 135 A/s.

c) The maximum current that will flow in the circuit can be calculated using the equation i_max = emf / R. Substituting the given values, we have:

i_max = 9V / 3Ω = 3A

Therefore, the maximum current that will flow in the circuit is 3A.

d) To find the expression for the current i(t) in the circuit as a function of time t, for t > 0, we can use the equation i(t) = (emf/R)(1 - e^(-t/τ)). Substituting the given values, we have:

i(t) = (9V / 3Ω)(1 - e^(-t/0.0667s))

Simplifying further, we get:

i(t) = 3(1 - e^(-15t))

Hence, the expression for the current i(t) in the circuit as a function of time t, for t > 0, is 3(1 - e^(-15t)).

Note: It's important to note that this answer assumes the switch is closed at t = 0 and there are no additional elements or factors affecting the circuit.