To find the speed of an electron accelerated to a kinetic energy of 50 GeV in the Stanford Linear Collider, we can use principles from relativistic physics. The mass-energy equivalence and relativistic momentum will help us determine the speed as a fraction of the speed of light (c) and the difference from c.
Understanding the Basics
First, let's clarify some important concepts. The rest mass energy of the electron is given by:
mc² = 0.511 MeV = 0.511 × 10⁻³ GeV
When the electron is accelerated to a kinetic energy of 50 GeV, its total energy (E) can be expressed as:
E = K.E. + mc²
Substituting the values, we have:
E = 50 GeV + 0.511 × 10⁻³ GeV ≈ 50 GeV
For practical purposes, the rest mass energy is negligible compared to the kinetic energy at this scale.
Relativistic Energy-Momentum Relation
In relativistic physics, the total energy of a particle is related to its momentum and mass by the equation:
E² = (pc)² + (mc²)²
Where:
- E is the total energy
- p is the relativistic momentum
- c is the speed of light
- m is the rest mass of the electron
Calculating Momentum
We can rearrange the equation to find the momentum:
p = √(E² - (mc²)²) / c
Substituting the values:
p = √((50 GeV)² - (0.511 × 10⁻³ GeV)²) / c
Since the rest mass energy is very small compared to the kinetic energy, we can approximate:
p ≈ 50 GeV / c
Finding Speed as a Fraction of c
Using the relationship between momentum, velocity, and mass, we have:
p = γmv
Where γ (gamma) is the Lorentz factor:
γ = 1 / √(1 - (v²/c²))
Substituting for p, we get:
50 GeV / c = γmv
Now, we can express v in terms of c:
v = pc / (γm)
Since γ = E / (mc²), we can substitute this into our equation:
v = pc / (E / (mc²))
After simplifying, we find:
v = pc² / E
Substituting p and E:
v = (50 GeV / c) c² / 50 GeV = c
Thus, the speed of the electron is:
(a) v/c ≈ 1
Calculating the Difference from c
To find the difference from c, we can use the Lorentz factor:
v = c(1 - 1/γ)
Where:
γ = E / (mc²) = 50 GeV / 0.511 MeV ≈ 9.8 × 10²
Now, substituting γ into the equation:
v = c(1 - 1/9.8 × 10²)
Calculating this gives:
v ≈ c(1 - 0.00102) ≈ 0.999 c
Thus, the difference from c is:
(b) c - v ≈ c - 0.999 c = 0.001 c
Summary of Results
In summary, for an electron accelerated to a kinetic energy of 50 GeV:
- The speed as a fraction of c is approximately 1.
- The difference from c is approximately 0.001 c.
This illustrates the fascinating nature of relativistic speeds, where particles can approach the speed of light but never quite reach it due to the effects of relativity.
