Q: Does an electric field have mass? Does it take energy to move an electric field?

The original question was: An electric field stores energy.  Energy has mass if I understand E=mc² correctly.

Now imagine a lone electron. It has an electric field. And therefore that field has mass presumably. If I apply a force to that electron, it will accelerate according to F=ma. My question relates to the m in F=ma.

The electric field must still exist even when the electron is moving. So therefore I am ALSO “moving” the electric field as well as the electron. So the m in F=ma must be made up of two parts, one of which is the mass component of the electric field and one of which just relates to the electron itself? Is that correct or am I confused. PS I appreciate it will be incredibly small and I also appreciate there may also be a magnetic field due to the changing electric field.

Physicist: You’re exactly right.  The electric field has mass (or, at the very least you could say that it has inertia and attracts things gravitationally), because it carries energy.  The energy density, K, of the electric field around a charge, q, is (ignoring all the physical constants for simplicity).  Near the charge (R=0) this equation doesn’t quite work, because the electron isn’t a point, but otherwise it holds up.

You can think of the energy in the field like a mess of Jello™ that’s thick near zero then thins out in all directions.  If you push the charge in the middle, the Jello™ will also move, but the movement will take the form of a jiggly wave that propagates outward.  That wave is where all the extra energy goes.

Electromagnetic energy.

Dropping the metaphor; pushing on a charge generates an electromagnetic (EM) wave.  So applying a force to something with a charge (like electrons) takes more energy than it should (based on the mass alone), because the act of pushing on it generates a spray of photons (which is light, which is EM waves).