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Suppose I need to find out the efficiency of electromagnet (solenoid) mounted on the table.

I energize it so that $1\text{lb}$ steel object gets attached to it. I energize it with DC current from a battery so that object only barely kept attached but does not press against the e-magnet, in other words the power provided to e-magnet produces only equal to the gravitational force of the object, no more. Assuming that the vectors of forces are fully vertical, force developed by e-magnet is equal to the weight of the object.

To calculate efficiency I know the input power $V\cdot I$ = $36\ \text W$ supplied to e-magnet but how to get the mechanical power required to keep the object attached up to e-magnet?

In larger retrospect I want to know how to find the mechanical power required to maintain object's potential energy in a static position without motion.

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If your goal is to maintain an object's position, then no power is required. Put an object on a table or hang it from the ceiling by a string and it will maintain its height above the floor indefinitely with no power input. You only need power if you want to move an object.

In your magnet setup, it is not the electrical power that is holding up the steel object, but the magnetic field created by the electrical current. If you used wire with a lower electrical resistance, less power would be required to maintain the same current. In fact, if you used superconducting wire, then no power would be needed since the current would persist without need of a battery. This is actually done for electromagnets in the Large Hadron Collider. Without electrical resistance, no power is lost, so no power input is required.

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  • $\begingroup$ @ Mark H If my goal were to keep object on the table I would not create this post. 2nd no current can be created without transformation of energy one way or another. 3rd Magnetic field which is holding object is created by energy spent in the form of current flowing through the coil. Resistance of the wire does not effect the required power at all. Strength of the field is B=uNI/L where current "I" must be generated with whatever wire providing needed voltage. Guess what will happen if coil is un-powered? Sorry but you need to go back to the textbook $\endgroup$ Dec 13 '19 at 3:29
  • $\begingroup$ Seems you assume that superconductor somehow creates electricity? The potential difference makes electricity and it's generation is outside of the superconductor discussed in your provided link. And surely energy is used to create it. Superconductor has only advantage of minimal losses, nothing to do with my question here. $\endgroup$ Dec 13 '19 at 3:42
  • $\begingroup$ @VladBlanshey (1) The magnetic field only depends on current, so if, for example, you use a lower resistance wire, you can use a lower voltage source, and thus use less power. Maintaining an object's position does not require power in general, so you can always reduce the power needed by changing the mechanism. (2) Inside a superconducting coil of wire, current can be maintained without a voltage source. Once a current is established, the coil can be short-circuited and the voltage source removed, so no more power is required. See the "Persistent mode" section in the link in my answer. $\endgroup$
    – Mark H
    Dec 13 '19 at 5:12
  • $\begingroup$ @VladBlanshey Another way to reduce or eliminate the required power is to insert an iron (or other ferromagnetic material) into your solenoid. The iron will become magnetized in response to current applied and add to the magnetic field of the solenoid, reducing the amount of current required. It may even be possible that the iron will acquire such a strong magnetic field that the object can be held up even after the electrical current is turned off. $\endgroup$
    – Mark H
    Dec 13 '19 at 5:17
  • $\begingroup$ 1. To maintain object position in this case requires force equal to the force of gravity. There are no known methods in science to create force without spending energy, hence no power used, no force created, no problem solved. 2. about usage of superconductor: to use super conductor it is not enough to create it but use a very power-hungry special power supply (read article why) and also maintain superconductor in subzero temps which requires energy far more than used in cheap power supply I can use here, 3. iron core is always used without saying when solenoid is used by engineer $\endgroup$ Dec 14 '19 at 20:47

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