Derivation of rate of energy change from Lorentz force

[Mcsheehy]

I am trying to derive from the Lorentz force equation the time derivative of the total energy. This involves using the equation for the jth electron in an electron beam traveling through an undulator. I have done it in such a way using the work done relation however I have been told that it is possible to derive it directly from the equation itself using:γ^-2=1-v²/c², d/dt(γm_ov)=-e(E+vxB), and the releation ½d/dt(v²)=v.d/dt(v).

The above were given as hints to derive the formula, but as far i can get is dot both sides with v and use the chain rule allowing me to sub in the γ term, but this leaves me with d/dt(ym_oc²) - d/dt(y^-1m_oc²) on the left and -2eE.v on the right. Not sure if I'm missing something. Any help would be great.

 

[mark726]

Hello,

Thank you for your question. It is definitely possible to derive the time derivative of total energy from the Lorentz force equation. Let me walk you through the steps:

1. Start with the Lorentz force equation: F = q(E + v x B), where F is the force on the particle, q is the charge, E is the electric field, v is the velocity of the particle, and B is the magnetic field.

2. We know that the work done on a particle is equal to the change in its kinetic energy. So, we can write: W = ΔKE = F ∙ Δr, where W is the work done, ΔKE is the change in kinetic energy, and Δr is the displacement of the particle.

3. Using vector calculus, we can rewrite this as: W = ∫ F ∙ dr, where the integral is taken along the path of the particle.

4. Now, let's substitute the Lorentz force equation into this expression: W = ∫ q(E + v x B) ∙ dr.

5. We can rearrange this equation to get: W = q ∫ E ∙ dr + q ∫ v x B ∙ dr.

6. The first term in this expression is the work done by the electric field, which we can write as: W = q ∫ E ∙ dr = q ∫ E ∙ v dt.

7. The second term is the work done by the magnetic field, which we can write as: W = q ∫ v x B ∙ dr = q ∫ v ∙ (B x dr).

8. We can use the fact that B x dr = dB, where B is the magnetic field and dB is the change in magnetic field along the path of the particle. So, we can rewrite the second term as: W = q ∫ v ∙ dB.

9. Now, we can use the chain rule to write: W = q ∫ dB/dt ∙ dt/dt = q ∫ dB/dt ∙ v dt.

10. Putting all of this together, we get: W = q ∫ E ∙ v dt + q ∫ dB/dt ∙ v dt.

11. We know that the time derivative of the total energy is equal to the work done on the particle, so we can write: dE/dt = q ∫ E