13. Relativistic Mass
In this section we will calculate the mass in relation to high velocities. The result is a simple isotropic increase in mass given by m=m0g2. But since the Lorentz force is also changed it makes a deflected particle appear to have the mass as described in relativity.
We found before that the mass was related to the self inductance and thus the change in energy of the magnetic field. Looking at Eq.(9-3) again, we can see that the magnetic energy of a charge is dependent on the component of electric field perpendicular to the velocity squared. That is, the change in magnetic field at relativistic velocity can be given by:
(1)
Note that, since only the perpendicular component of electric field contributes, Eq.(12-1) is used with q = p /2. So the relativistic change in magnetic field energy can be compared to the non-relativistic energy by:
(2)
As this is a classical theory the basic law of kinetic energy is the same at all velocities provided that the change of velocity is small. Therefore we can write
(3)
and thus
(4)
This represents a relativistic increase in mass. It is not a question of perspective, but a real mass increase relating to the absolute velocity of a charge (or mass). This classical mass increase is isotropic in respect to direction.
Imagine that one uses the classic method of determining the mass of a particle. That is by deflecting it in a magnetic field and an electric field. The equations normally used would be
(5)
(6)
(where v is now the relative velocity) and the centripetal force:
(7)
We can substitute Eq.(7) into (6) and (5) with R the radius of the particle's path in the electric field and r the radius of the particle's path in the magnetic field. This yields:
(8)
(9)
Solving simultaneously for the velocity and mass results in:
(10)
(11)
but in reality, both fields have an increased force (Eq.(12-15)). Since the determination of velocity is a simple ratio of the magnetic field and the electric field, the result is unchanged and the same velocity is derived. However, the magnetic field is squared in determining the mass, so the mass would be incorrectly determined such that:
(12)
where m' is the apparent mass and m is the actual mass. If the mass of a particle were determined by deflection and one assumed that the force on the particle were independent of velocity, the apparent mass increase1 of the particle would be, by using the factor of Eq.(12) on Eq.(4):
(13)
1. This compares with observations, some classic examples being Rogers, McReynolds and Rogers3 using electrons; Zrelov, Tapkin and Farago4 using protons.
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