the switch is closed at t=0 s. Find energy stored in the inductor at t=τ (where τ is time constant of circuit) (see attachment for diagram)

To determine the energy stored in the inductor at time τ (the time constant of the circuit), we first need to understand how inductors behave in an RL circuit. When the switch is closed at t=0 s, the inductor begins to store energy as the current through it increases. The energy stored in an inductor can be calculated using the formula:

Energy Stored in an Inductor

The energy (W) stored in an inductor is given by the equation:

W = (1/2) L I²

where:

• W = energy in joules (J)
• L = inductance in henries (H)
• I = current in amperes (A)

Current in the Circuit

In an RL circuit, the current as a function of time after the switch is closed can be expressed as:

I(t) = (V/R) (1 - e^(-t/τ))

where:

• V = voltage across the circuit
• R = resistance in ohms (Ω)
• τ = L/R, the time constant of the circuit

Evaluating at Time τ

At time τ, we can substitute τ into the current equation:

I(τ) = (V/R) (1 - e^(-1))

Since e^(-1) is approximately 0.3679, we can simplify this to:

I(τ) ≈ (V/R) (1 - 0.3679) = (V/R) (0.6321)

Calculating Energy at Time τ

Now, we can substitute I(τ) back into the energy formula:

W = (1/2) L [(V/R) (0.6321)]²

Expanding this gives:

W = (1/2) L (V²/R²) (0.6321)²

Calculating (0.6321)² yields approximately 0.3989, so we can write:

W ≈ (1/2) L (V²/R²) (0.3989)

Final Expression for Energy

Thus, the energy stored in the inductor at time τ can be expressed as:

W ≈ 0.19945 (L V²/R²)

This formula provides a clear way to calculate the energy stored in the inductor at the time constant τ, based on the inductance, voltage, and resistance in the circuit.