Electronic – specifying a non-momentary switch

It is not really clear what you are trying to do. If I ignore the bit about "current flowing" then it looks as if you want a signal that indicates there is motion of the "motor." Are you actually trying to detect a motor moving, or some other object? Different methods can be applied in either cases.

Inferring that you are trying to detect that some object is moving in a repeatable fashion and you need to do so without physical contact with that object, my suggestion is attach an object that can be detected to the moving part and put a sensor on the fixed part.

If the moving part is indeed a VCR cassette, then a magnet is a bad idea as it will affect the recording on the tape. So let's use a reflector on the moving bit and a reflective optical sensor on the fixed part. The sensor looks like this

Similar sensors can be purchased here. They combine an infrared emitter and a photodiode. The light goes out the emitter, reflects off the sensor and is detected by the diode.

That sensor will produce a pulse on the output when the reflector is seen. To extend the length of the pulse so it lasts for a full rotation before being triggered again, a circuit called a one-shot multivibrator can be used

This circuit will provide a constant signal as long as the rotation continues. After one full period without the rotating sensor being detected, the signal will shut off.

Hope that helped.

This does not use a relay but a FET, seems to tolerate your voltage needs (from 4.5V to 20V). You have to check if 10A max is ok for your application.

http://www.pololu.com/product/750

If you bypass the push button with your momentary switch you should have what you need.