# Electron in varying magnetic field

Lets consider an electron that is placed in an existing, constant in space near the electron, magnetic field. Electron is stationary. Magnetic field over time gradually reduces to zero.

I assume electron will accelerate. How can I calculate how much will it accelerate?

Usually I find effects on moving electrons and static magnetic fields. i assume I can also take the reference frame of an electron initial trajectory, to get these common cases to the case im interested in. Can I do that? Is there some other simple way to calculate it approximately, without integrals?

If you think many solutions exist, consider the one that is closer to a situation of a loop of wire with current and electron in the middle of this loop. Current in the wire makes the magnetic field. Changing current in the wire, probably, makes electric field. Electron makes electric field. Electron that is probably accelerating would make magnetic field due to motion. No other fields exist: if wire loop and electron are removed, then space will quickly calm down to zero electric and magnetic field. Point of interest, where fields are measured is the position of the electron, in the middle of the loop. Time over which system evolves is small enough that electron, if accelerated, doesnt move far from its original position. Relativistic effects, if possible, to be ignored, so that wavelength associated with the rate of change of magnetic field is significantly larger than the wire loop size. But simpler solution is preferred if several solutions allow to get an answer for this case with reasonable precision - within x0.5...x2.

• Do you mean constant in all space or constant over a cylindrical region?
– hft
Apr 11 at 20:04
• @hft Im interested in a case of a wire loop. It makes a field that is somewhat uniform nearby the electron in the center. But i dont want these details to make everything complicated, simpler solution is better Apr 11 at 20:07
• Does this answer your question? What´s the electric field in the entire space generated by a uniform (but not constant) magnetic field?
– hft
Apr 11 at 20:15
• Then you should edit your question to state the actual question you have. As it is currently posed (truly constant in all space) there is no way to answer.
– hft
Apr 11 at 20:16
• @hft i dont agree with the answer in the link. It seem person that answers assumed electric field is constant and zero, while user asked what the electric field will be. And answer says electric field will be zero, while intuition says it wont be - electron will accelerate from static position. Probably some details in the way question is formulated made the answer non-intuitive. Apr 11 at 20:24

For example, if you have a uniform field over a circular area of radius $$R$$ and the electron at distance $$r from the center, the magnitude of electric field at the electron will be $$E=\frac{1}{2}\dot Br$$ where $$\dot B$$ is the rate of change of magnetic field with respect to time. The direction of the field is tangent to the circle of radius $$r$$ and given by the right hand rule.