Magnetic fields are difficult to draw. They are invisible and three dimensional.
I'm not going to worry too much about the field, except to I know that it contains energy. The energy in the field increases with the square of the current in the coil. I can describe the mount of energy that a coil will store with a number which is called the inductance of the coil. The formula for this is:
This means that you only need to know the inductance of the coil and the amount of current flowing through the coil to figure out how much energy is stored in it.
There are more formulas that can be used to calculate inductance from things like the number of turns in the coil, the size of the coil, and the size of the wire. There are books that have explain these formulas. Inductance is such a nice way to describe coils that coils are often called "inductors."
Now, suppose that yo have a nice big magnetic field because you built an electromagnet and hooked it up to a battery. Then you disconnect the battery and the magnetic field disappears. Where did the energy that was stored in the field go?
Much of the energy goes into a little spark that jumps across the spot where the wire gets disconnected. Some clever things can be built by using this spark for something. For instance, you can make a circuit that transforms a low voltage from a battery into a higher voltage.