# Back emf when circuit is closed and opened and the conservation of energy

Ok, this might sound like a stupid question but I am legitimately confused. Imagine a circuit connected to a solenoid. When a circuit is closed (by a switch), there will be a quick increase in current, which will induce a magnetic field in the solenoid. The same magnetic field causes a change in the magnetic flux linkage of the coils, which then produces a back emf. Supposedly, this emf opposes the battery so the law of conservation of energy is not violated (my guess is that the kinetic energy supplied to the electrons by the battery is the source of the back emf?). However, when the circuit is opened, the same thing happens, but this time the emf is in the direction of the battery. I don't understand how this doesn't violate the law. As I see it, there is no energy in the circuit to be "transformed" into the back emf. Some texts I read said that opening the circuit is akin to an emf in the opposite direction as current suddenly drops, but there isn't any actual work done when opening the circuit?

If anyone can shed some light on the origin of the "energy" fueling this back emf, I will be very grateful.

## 1 Answer

Don't worry, electromagnetism is a bit of a handful at first. The best way, I found, to understand it is to always go back to Maxwell's equations.

"When a circuit is closed (by a switch), there will be a quick increase in current, which will induce a magnetic field in the solenoid. The same magnetic field causes a change in the magnetic flux linkage of the coils, which then produces a back emf. "

Assuming your circuit is a simple ideal solenoid and voltage source, the increase in current is linear and unbounded. This is an important point because it is the change in magnetic field that induces the back emf. Therefore your point:

"my guess is that the kinetic energy supplied to the electrons by the battery is the source of the back emf?"

is incorrect. It is the change of the magnetic field that is the source of the back emf, per Maxwell's equations.

Now, we're linearly increasing the current in the solenoid and you decide to open the circuit. What happens? The current wants to continue flowing and will force itself through the air gap to close the circuit.

"but this time the emf is in the direction of the battery. I don't understand how this doesn't violate the law. As I see it, there is no energy in the circuit to be "transformed" into the back emf."

The energy comes from the magnetic (and electric) field which also stores energy.

• I think I'm starting to understand. However, I am still unsure of how the magnetic and electric field "stores" energy. So when current is flowing normally, is a fraction of the energy used to generate a magnetic field? – chematwork Feb 4 at 14:21
• Unless there's a resistance or capacitance in your circuit, all the energy is used to set up the magnetic field. Consider an ideal V source shorted with a 0 Ohm impedance. The current will quickly increase linearly, the rate determined by the inductance of the circuit (never mind that). Your V source is certainly putting out power. Where does the energy go? To the magnetic field set up by the flowing current! Incidentally, the energy stored is proportional to the inductance of the circuit. – user16035 Feb 4 at 17:43
• In general all fields store energy. For E&M if you're comfortable that a capacitor stores energy, that energy is "in" the electric field. You preformed work to set it up. Now, if you're comfortable that electric fields store energy, than you must accept that magnetic ones do too (first because you've accepted that a field can store energy, but also because the electric field and the magnetic field are actually the same thing, just in a different inertial frame! see your text's section on special relativity, but it's actually a quite simple thought experiment). – user16035 Feb 4 at 17:44
• One last comment, perhaps you feel weird about shorting a perfect V source with a 0 Ohm resistor because someone told you that that causes infinite current. They lied to you the current isn't infinite, but the circuit theory approximation has broken down. All circuits must have an inductance. This inductance limits the current to an unbounded, linearly increasing amount. No matter what, if you're in doubt, always go back to Maxwell. If anything ever feels weird, first review your understanding of vector math, then review your understanding of Maxwell. – user16035 Feb 4 at 17:45