Consider the RL circuit shown in the figure, where R = 5.0 Ω and L = 4.6 H. How much energy is stored in the inductor after the switch has been closed a long time? I would know how to do this if they actually gave me a value for time, but I'm lost now. Can someone help me find the right equation to use?

I assume that the resistor and the coil are in series

I assume you are given the voltage, V

In that case, after a long time the current through the inductance is CONSTANT and there is NO voltage across the inductor
E = L di/dt but di/dt is zero
so only the resistor matters for i
i = V/R
then
energy = (1/2) L i^2

To determine the energy stored in the inductor after the switch has been closed for a long time, you can use the formula for the energy stored in an inductor, which is given by the equation:

E = (1/2) * L * I^2

Where:
E is the energy stored in the inductor (in joules)
L is the inductance of the inductor (in henries)
I is the current flowing through the inductor (in amperes)

In this case, since the switch has been closed for a long time, we can assume that the current through the inductor has reached its steady-state value, which is given by Ohm's law:

I = V/R

Where:
V is the voltage across the resistor (in volts)
R is the resistance of the resistor (in ohms)

Since the circuit shown in the figure is an RL circuit, when the switch is closed for a long time, the inductor reaches its steady-state condition, and the voltage across the resistor is equal to the applied voltage. Therefore, we can replace V with the applied voltage (let's call it V_applied) in the equation.

Now, to find the energy stored in the inductor after the switch has been closed for a long time, we need to find the current flowing through the inductor using Ohm's law:

I = V_applied / R

Once we have the current value, we can substitute it back into the equation for the energy stored in the inductor:

E = (1/2) * L * (V_applied / R)^2

Since you have been provided with the values for R (5.0 Ω) and L (4.6 H), you can find the energy stored in the inductor by plugging in these values into the equation along with the applied voltage (V_applied) if it is given.