Why does this work? Right hand rule
IF we just keep this position, the motor will rotate back and forth, i.e. not uni-directionally. This occurs because at some point in the revolution, the direction of torque will flip. So we use a commutator to switch the currents so this doesn't happen!
This makes a torque!
Force on one wire on one side: $F=IlB$ Force per coil, given $N$ turns of a coil: $F=NIlB$ Hence, as $\tau=rF\sin \theta$, $\tau = rNIlB\sin \theta$ - One side!!!
Hence, if we consider the other side: $\tau_{\text{max}}=2NIlBr\sin \theta$
This is the maximum torque on a coil in a motor.
But consider that $2lr$ is essentially $ld$ where d is the diameter. Hence, we can represent this as $A$, i.e. the area of the shape of the coil.
Hence, $\tau_{\text{max}}=NIAB\sin \theta$
Note that torque will change over time.
