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Lorentz factor vs Energy for Electrons and Protons

Lafky
Apr 7, 2021
1 min read

A common question asked of light sources is, "why electrons"?

After all, if any charged particle emits radiation when moving through a magnetic field, then surely protons would be just as useful?

Actually no. Larmor's formula tells us the power emitted by a charged particle is proportional to the Lorentz factor raised to the 6th power!

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In practice when the acceleration is orthogonal to the velocity, as must be the case with magnetic fields, two of the gammas cancel out leaving you with P ∝ γ^4.

We can see that for a given energy of 3 GeV, electrons have γ ≈ 5871 whereas protons have a γ ≈ 4.2. The ratio of power radiated then is on the order of 3.8e12.

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