Problem on Relativistic Momentum and Energy
Recall an effect of photoelectricity discussed in Waves - Photoelectricity chapter of UNIZOR.COM course Physics 4 Teens - a ray of light, directed towards a metal plate, kicks electrons from a metal surface.
Electrons, kicked out of metal by a ray of light, have energy and momentum.
Therefore, according to the laws of conservation of energy and momentum, light must carry energy and momentum.
In the previous two lectures of this chapter of the course we have derived formulas for relativistic momentum p and relativistic kinetic energy K of an object. These formulas depend on the object's rest mass m0 and its speed u:
| p = |
|
= γ·m0·u |
| K = |
|
− m0·c² = |
where c is the speed of light in vacuum and γ is Lorentz factor
| γ = |
|
The problem to apply these formulas to light is that, on one hand, light has zero rest mass m0=0 and, on the other hand, Lorentz factor for light (u=c) is not defined since it turns into division by zero.
In spite of all these difficulties, here is a problem.
Problem
What is the ratio of relativistic kinetic energy of light to its momentum K/p?
Solution
Obviously, we cannot substitute light characteristics m0=0 and u=c directly into formulas for energy and momentum because of undefined Lorentz factor.
Instead, let's consider an object of some rest mass m0 and very high speed u.
Then for this object
| K/p = (c²/u)· |
|
K/p = (c²/u)·(1−1/γ)
Now we can substitute parameters for light:
u = c
1/γ = √1−c²/c² = 0
The result of this substitution is
K/p = c
Answer
For light in vacuum the ratio of relativistic kinetic energy to its momentum is
K/p = c
Consequently, the momentum of light can be expressed in terms of its kinetic energy
p = K/c
Historical Note
The formula for ratio of energy of a photon to its momentum was derived from Maxwell equations long before the Theory of Relativity was invented by Einstein.
Compton effect and Poynting Theorem were the bases for this derivation.
We have derived this ratio from relativistic standpoint without directly resorting to Maxwell equations and properties of electromagnetic waves.
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