But that doesn't make sense to me because it's not a resistor; it's a
coil, more like an inductor.
It's not an ideal resistor - since ideal resistors have only resistance - and it's not an ideal inductor -since ideal inductors have only inductance.
If this were a loop of ideal conductor, which has zero resistance, a constant current could exist in the loop without an emf generating, time changing magnetic field linking the loop since there is no dissipation of energy.
However, when there is resistance in the loop, sustaining a current $I$ requires a non-zero emf since the resistance dissipates energy.
When the voltage across the resistance (given by Ohm's law) and the emf generated by the time changing magnetic field are of the same magnitude, the current is constant with time.
Does that mean energy is stored in an electric field produced by a
point charge?
The classical self energy of a point charge is formally infinite and, thus, somewhat of an embarrassment.
Could you explain how energy can be stored in a magnetic field?
It's clear that energy is stored in a magnetic field so I'm not sure what you're looking for here.
When work is done, energy is converted from one form to another. Work is being done by the battery when the current through an inductor is increasing.
This is simply due to the fact the product of voltage and current is power, the rate at which work is done.
And, when the magnetic field threading the inductor coils is changing, there is a voltage across the inductor.
Thus, when the current through the inductor is increasing, there is a voltage across, proportional to the rate of change of current, and thus, an associated power
$$p_L = i_L \cdot v_L = i_L \cdot L \frac{di_L}{dt} $$
Further, when the current is decreasing, work is being done by the inductor on the circuit. So, the work done increasing the current (and associated magnetic field) is stored as energy in the magnetic field.
Best Answer
Like Peter Diehr says in the comments, the way to see the duality between inductors and capacitors is that capacitors store energy in an electric field, inductors store energy in a magnetic field.
No, the magnetic field is proportional to the current, so if you stop the current then the field will go to zero. But because the stored energy is proportional to the current, you actually can't stop the current without doing something to remove the stored energy.
In duality to how a capacitor can store energy when no current is passing through it, and inductor can continue to pass a current (and thus store energy) when the potetntial difference across it goes to zero.
To stop the current, you have to apply a potential opposing the established current flow, which will mean the inductor delivers energy to the rest of the circuit.