# Angular Velocity of Electrically Powered Motor

1. Aug 14, 2015

### Physicist97

Hello! This isn't a homework question, and I don't think it is too homework-like, but if I'm mistaken I apologize. My question is if you had a battery, or some source of electrical energy, hooked up to a coil of wire in a constant magnetic field, in such a way that the wire spins around (basically an electric motor), would this be a correct equation for angular velocity of the motor? (I have heard electric motors have something called a Commutator on them so that even a direct current will switch directions periodically, so let's assume this is part of my hypothetical motor).

The power from the battery will be $P=V^{2}/R$ where $V$ is the voltage of the battery, $R$ is the total resistance. Power is also the inner product of the torque and angular velocity $P={\tau}{\cdot}{\omega}$. The magnitude of torque produced by a current in a magnetic field I looked up in my notes as ${\tau}=N(V/R)ABsin({\theta})$ , where ${\theta}$ is the angle between the a unit normal vector of the area of the loop, $A$, and the magnitude of the magnetic field, $B$, $N$ is the number of turns of wire for the loop and $V/R$ is equal to the current going through it. So plugging that torque into the definition of power gives you $V^{2}/R=N(V/R)ABsin({\theta})n{\cdot}{\omega}$ , where $n$ is a vector pointing in the direction of torque. Simplifying and solving for ${\omega}$ gives $n{\cdot}{\omega}={\frac{V}{NAB}}csc{\theta}$.

Is this correct, or have I made a mistake? Thank you!

2. Aug 14, 2015

### Staff: Mentor

RI=V is no longer valid as the motion of the coil in the field will contribute to the voltage.
Also, $\omega$ is not constant as torque depends on the orientation of the coil.

3. Aug 14, 2015

### Physicist97

Thank you for the quick reply! Right, due to Faraday's Law an electric potential is generated, completely forgot about that. Would it be possible to figure an equation like this out or will it just become more and more complicated?

4. Aug 14, 2015

### Staff: Mentor

If you include induction, you get a torque that depends on both the angle and the angular velocity. This leads to a differential equation. If you also add some friction, it might be possible to average over one cycle, then the equation should have a nice solution.

5. Aug 15, 2015

### stedwards

****There are many motors-- things that translate electrical energy to mechanical energy. Which one do you have in mind?

6. Aug 15, 2015