Why the left hand rule?

Today’s post is about the magnetic field and what it really means in the context of the force felt by a charge moving through it. From high school physics, we know that a moving charge (0r current) in a magnetic field feels a certain force on it which is perpendicular both to the direction of motion and the direction of the magnetic field. To find out the direction of this force, we can either follow through with the vector cross product of the current and the magnetic field or use one of the many hand tricks like the left hand rule below.

Fig: 1 Left hand rule

The first question that comes to my mind really is, what is magnetic field? A moving charge as well as the effects of force in motion are both observables. But, how do we observe the magnetic field? If we don’t, how do we know it is there? The magnetic field, like other fields, is a creation of science, not nature. We use it to help explain observables like current and acceleration.

Let’s assume there are magnets as shown in Fig:1, between which is a conductor carrying current in the direction shown. Then, we observe a force on the conductor in the upward direction. We observe that when the magnets are removed from the picture, there is no longer a force on the conductor. Also, when one of the magnets is flipped so that they are repelling, there is no force on the conductor. But, when both the magnets are flipped so that they attract again, we see that the force on the conductor is now of the same intensity as the first case, but downwards. If we are to define two poles for a magnet- north and south and imaginary lines emanating from the north pole and sinking into the south pole, we see that there are such lines crossing the conductor only when there is a north pole and a south pole on either side of it and this, we might hypothesise, encapsulates the effect of the magnet on the conductor. So, the idea of the magnetic field (it it works) replaces the whole structure of the magnets with one space and time variant vector, as far as the current carrying conductor is concerned.

As myriad experiments prove the utility of the magnetic field idea and validates our left hand rule, we still don’t know if this is the best method to reduce the effects of the magnets. We see that this definition of magnetic field is perpendicular both to the current and the direction of force, the two directions where there are observable effects. Why not define the magnetic field in the direction of the force like the electric field or gravitational field is? To explore why, let us use the method of contradictions. Assuming the direction of magnetic field is defined along the force on the conductor, we see that the current is perpendicular to both the magnetic field and the force which lie along the same axis. If the current is now flipped, the force, according to the previous rule will still be along the same direction as it remains perpendicular to current and parallel to the magnetic field. But, we observe that when the direction of current is flipped, so is the direction of the force on the conductor. So, we cannot define magnetic field in the direction of the force. We can also rule out defining it in the direction of the current because then, we do not have any easy way to judge the direction of force, apart from restricting it to a plane perpendicular to the current and field. But, if we do choose a direction orthogonal to both the current and force, we see that simple rules like the left hand rule can be developed to find one observable from the other in a way that agrees with experiment.

While all fields are human creations, I have always found the idea of the magnetic field the most artificial. I don’t have any rational arguments for this feeling, but it has to do with how it is defined in a direction which is apart from either of the observable vectors involved. But, as we see, this is the only way we see how to define the magnetic field in a way that facilitates accurate predictions between current and force.