Physicist97
- 31
- 4
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!
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!