# Geneva LHC (Homework 5, Physics 226)

(a) What gamma-factor will protons accelerated to the design energy have?

(b) What is the angular velocity !lab of these protons as measured by a stationary clock?

(c) What is the proper-time angular velocity !p of the protons?

(d) What is the magnitude of the 4-acceleration of the protons?

(e) Accelerating any charged particle causes it to emit electromagnetic radiation. The power radiated (i.e., the rate at which energy is lost via electromagnetic radiation) by an accelerating particle with charge q was rst derived by J. Larmor in 1897, and is given by $P_{rad} = (2k_e/3)q^2a^2/c^3$ where $a^2$ is the square of the 4-acceleration. At the design parameters of the LHC, what total power, in watts, will be radiated by the protons circulating around the LHC ring?

a)

Equation (3.3.5) is

$γ=E/(m_0 c^2 )$

For protons $m_0 c^2=938.272 MeV$ so that

$γ=(7*10^{12})/(938.272*10^6 )=7460.5$

b)

$γ^2=1/(1-v^2/c^2)$ so that $v^2/c^2 =1-1/γ^2$ and $v=c\sqrt{(1-1/γ^2)}$

$ω_{lab}=v/R=c/R \sqrt{(1-1/γ^2)}=(3*10^8)/((27*10^3)/2π)*\sqrt{(1-1/7460.5^2)}=$

$=1.3963*10^5 ( rad/s)$

c)

$ω=2π/T$ so that $T=2π/ω_lab =45.00 μs$

The Lorentz invariant is

$d s^2=-c^2 t^2+d x^2$ in laboratory and $d s^2=-c^2 τ^2+0$ (in rest system)

So that the proper time (time in rest reference system) is

$c^2 τ^2=c^2 T^2-d x^2=(3*10^8 )^2*(45*10^{-6})^2-(27*10^3)^2=-5.4675*10^8 (s^2)$

$τ^2=-1.8225 s^2$

$ω_p^2=(4π^2)/τ^2 =-21.66 (rad/s)^2$

d)

$p^2=-m^2 c^2$ and $f=p/τ$ so that $f^2=(-m_0^2 c^2)/τ^2$

$a^2=f^2/(m_0^2 )=-c^2/τ^2 =(3*10^8 )^2/1.8225=4.938*10^{16} (m^2/s^4 )$

e)

$P=2/3*k e^2*a^2/c^3 =2/3*9*10^9*(1.6*10^{-19} )^2*(4.938*10^{16})/(3*10^8)^3=$ $=2.81*10^{-37} W/proton$

Total number of protons is

$N=2800*10^{11}$ so that $P_{tot}=7.87*10^{-23} W$