# Energy in an Electric Field

I understand the notion that an electric field admits an electric potential field, giving the energy per unit charge at each point in space. But I don't understand what is meant when someone says the field itself has energy.

Can anyone explain the motivation behind assigning an energy to an electric field? How can this energy be transformed to other forms of energy? What experiments make use of this idea?

A really, really good motivator is that it's a lot easier to deal with scalar quantities such as potential energy over electric fields when calculating things such as forces.

A really, really good motivator is that it's a lot easier to deal with scalar quantities such as potential energy over electric fields when calculating things such as forces.

Again, I'm not talking about the electric potential of the field. I'm talking about the intrinsic energy of the field, given by the equation:

$$U_e=\int_{\tau}\frac{1}{2}\epsilon E^2 d\tau$$

So another way to phrase my question is where does this equation come from?

jtbell
Mentor
Can anyone explain the motivation behind assigning an energy to an electric field? How can this energy be transformed to other forms of energy? What experiments make use of this idea?

For one thing, electromagnetic waves transfer energy. If the energy doesn't reside in the E and B fields while it's in the process of being transferred, where is it?

Dale
Mentor
2021 Award
Can anyone explain the motivation behind assigning an energy to an electric field? How can this energy be transformed to other forms of energy? What experiments make use of this idea?
My favorite derivation is here: http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

As far as what experiments make use of the idea, it is well known in MRI that there is a huge amount of energy stored in the magnetic field. When a magnet quenches one of the primary challenges is to dissapate all of that energy safely. Luckily, you can dump a lot of energy into liquid nitrogen as long as you can vent the nitrogen somewhere where it will not asphyxiate people.

Since the post above contains the math (which I am going to have to ponder) I will present another more nonmathematical view that helps me understand your question.

If an electric field exists it can imply that a charged object exists somewhere that produced that field. If you tried to bring another object with the same charge into that field and move it closer and closer to the charge that created that field, it would require that you do work against that field. You would have to use a force to drag that new charge closer to the original charge. So by this reasoning an electric field automatically implies for me an imbalance. Imbalances of any type intrisically have potential energy in my world.

Now that equation looks very much like the energy density little u in some texts (energy per unit volume) that exists in a uniform electric field like between the plates of a capacitor. u = 1/2*epsilon*E^2. But since you are not necessarily looking at a uniform E field it would require summing up (the integral sign) with respect to the position within that electric field since it is not necessarily uniform. You have prodded me to look at the math in the post above a little closer as it is difficult for me to put the math with the idea sometimes. And maybe I am way off in my nonmathematical explaination or it needs to be cleaned up.