Question Video: Calculating the Charge-to-Mass Ratio of a Particle Moving through a Uniform Magnetic Field Physics

Video Transcript

A charged particle moves through a uniform magnetic field. It moves along a circular path with a radius of 0.0200 meters. Its speed is 3.40 times 10 to the sixth meters per second. The strength of the magnetic field is 0.200 teslas. What is the charge-to-mass ratio of the particle? Give your answer in coulombs per kilogram to three significant figures.

Since this question is asking us to find the charge-to-mass ratio of a particle moving through a uniform magnetic field, it makes sense to consider the equation speed divided by magnetic field strength times radius is equal to charge-to-mass ratio. This gives us the charge-to-mass ratio of a particle moving in a uniform magnetic field with motion described by the speed and the radius of the path. Recall that we derived this equation by equating the magnetic force on a charged particle moving in a magnetic field to the centripetal force on any object moving in a circle.

Now that we have this equation, to calculate charge-to-mass ratio, all we need to do is find appropriate values for the speed, magnetic field, and radius. In this question, the values are simply given to us. We have 0.0200 meters for the radius, 3.40 times 10 to the sixth meters per second for the speed, and 0.200 teslas for the strength of the magnetic field. Plugging those values into our formula gives us an expression for the charge-to-mass ratio. Let’s start by evaluating the numerical portion, 3.40 times 10 to the sixth divided by 0.200 times 0.0200. Plugging into a calculator, this gives us 850 times 10 to the sixth.

We’re asked to report our answer to three significant figures. Since this number is written in the form of three digits, 850 times a power of 10, the three significant figures for this number are 850. We could either leave the number in this form, or to be absolutely clear with the significant figures, we could write it as 8.50 times 10 to the eighth. For the units, meters in the numerator divided by meters in the denominator is just one. And per seconds in the numerator divided by teslas in the denominator is just one divided by tesla seconds.

Aside from the fact that these are somewhat odd units, the question wants us to have our answer in coulombs per kilogram. So we need to convert one per tesla seconds to coulombs per kilogram. The most direct way to do this is to recall that one tesla is exactly one kilogram per coulomb seconds. So one tesla second is one kilogram second per coulomb second, which is just one kilogram per coulomb. Taking the reciprocal of both sides, we see that despite one per tesla second looking rather odd, it’s exactly equivalent to one coulomb per kilogram, which is precisely the units we want for our answer.

This one-to-one relationship is actually guaranteed from the definition of SI units and the fact that the tesla is the basic unit of magnetic field, the second is the basic unit of time, the coulomb is the basic unit of charge, and the kilogram is the basic unit of mass. Because one per tesla seconds and coulombs per kilogram are both valid units for charge-to-mass ratio and both only contain basic SI units, by definition they must be equivalent. So to three significant figures, the charge-to-mass ratio of our particle is 8.50 times 10 to the eighth coulombs per kilogram. As it happens, this is almost exactly the magnitude of the charge-to-mass ratio of both the muon and the antimuon.