Invariant mass and energy balance

In summary, the invariant mass of a two-particle system is given by the following equation: E1 + E2 = X, where X is the invariant mass.
  • #1
R3ap3r42
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3
Homework Statement
This is a question from a Special Relativity course. Uses invariant mass, Laurentz Transformation
Relevant Equations
Invariant mass, Lorentz Transformation, Conservation of Energy, Conservation of Momentum
a) Two particles have energies E1 and E2, and momenta p1 and p2. Write down an expression for the invariant mass of this two-particle system. Leave your answer in terms of E1 and E2, and p1 and p2.

b) A typical photon (γ) in the Cosmic Microwave Background (CMB) has an energy of kBTCMB, where TCMB = 2.73 K and kB is the Boltzmann constant. Such a photon can collide with a high-energy proton via the reaction p + γ → ∆+, where the ∆+ particle has a mass of 1.23 GeV/c2 .

i) Calculate the energy of the CMB photon in eV. [2 marks]

ii) If the proton and photon collide head-on, show that their invariant mass, $$ M_{inv} $$, satisfies

$$ M_{inv}^2 c^4 = m_p^2 c^4 + 2k_B T_{CMB} (E + cp) $$

where E, p and mp are the proton’s energy, momentum and mass. [4 marks] Hence show that the proton energy can be written

$$ E= \frac {m_∆^2 c^4−m_p^2 c^4} {4k_BT_{CMB}}+δE $$ and determine δE in terms of the particle masses and the photon energy.

iii) Compute the numerical values of δE and E in eV. [2 marks] iv) How would your expression for the proton energy change if the photon and proton collided at right angles?I got all the way to b) ii but I could not get to the given expression for E.
Can anyone point give me any clues? I am pretty sure it's just algebraic work that I can seem to simplify.

I manage to get to this:

$$ E= \frac {m_∆^2 c^4−m_p^2 c^4} {2k_BT_{CMB}} - cp $$

This seems close but I can't get rid of the cp (also note that mine is divided by 2 not 4 as expected).

Thanks a lot.
 
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  • #2
R3ap3r42 said:
Homework Statement:: This is a question from a Special Relativity course. Uses invariant mass, Laurentz Transformation
Relevant Equations:: Invariant mass, Lorentz Transformation, Conservation of Energy, Conservation of Momentum

I got all the way to b) ii but I could not get to the given expression for E.
Show your work please.
 
  • #3
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Hope it is clear enough.
 
  • #4
R3ap3r42 said:
I manage to get to this:

$$ E= \frac {m_∆^2 c^4−m_p^2 c^4} {2k_BT_{CMB}} - cp $$

This seems close but I can't get rid of the cp (also note that mine is divided by 2 not 4 as expected).

Thanks a lot.
You're missing the same trick I showed you yesterday. You have:
$$E + pc = X \ \ (1)$$where$$X = \frac {m_∆^2 c^4−m_p^2 c^4} {2k_BT_{CMB}}$$Now$$m_p^2c^4 = E^2 - p^2c^2 = (E-pc)(E+pc) = (E-pc)X$$$$\Rightarrow \ E - pc = \frac{m_p^2c^4}{X} \ \ (2)$$Now, add equations (1) and (2).
 
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Wow! This is unbelievable. I was blind and now I see. :)
I guess a need to do 100 more of these.

Thanks a lot!
 

FAQ: Invariant mass and energy balance

What is invariant mass?

Invariant mass is a property of a system of particles that remains constant regardless of the reference frame in which it is observed. It is a fundamental concept in physics and is related to the total energy and momentum of the system.

How is invariant mass calculated?

Invariant mass is calculated using the equation E^2 = (pc)^2 + (mc^2)^2, where E is the total energy of the system, p is the momentum, c is the speed of light, and m is the rest mass of the system.

What is the relationship between invariant mass and energy balance?

Invariant mass and energy balance are closely related. Invariant mass represents the total energy of a system, including both its rest energy and kinetic energy. Energy balance refers to the conservation of energy in a system, which is based on the principle of conservation of invariant mass.

How is invariant mass used in particle physics?

Invariant mass is a crucial concept in particle physics, as it allows scientists to identify and study different types of particles. By measuring the invariant mass of particles produced in high-energy collisions, scientists can determine their properties and interactions.

What are some real-world applications of invariant mass and energy balance?

Invariant mass and energy balance have many practical applications, such as in medical imaging techniques like positron emission tomography (PET) scans, which use the principles of invariant mass to create images of the body. They are also used in nuclear reactors and in the development of new energy sources.

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