taha.hojati
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Homework Statement
A point charge q is moving relativistically with constant speed ##\beta## along the x-axis.
At t = 0 a constant decelerating force F is applied in opposite direction of
its velocity. If the charge stops after traveling a distance d, find the total radiated
energy.
Homework Equations
I found these equations in Griffith's book:
Total radiated power by a relativistically moving point charge (Lienard's eq.):
$$P = \frac{\mu_0 q^2 \gamma^6}{6 \pi c} (a^2 - |\frac{\beta \times a }{c} | ) $$ Here again ##\beta## is the the velocity of the charge (in SI units), and a is its acceleration.
Newton's law of motion for charged particle (with radiation reaction):
$$a= \tau \dot{a} + \frac{F}{m}$$ here ##\tau = \frac{\mu_0 q^2}{6 \pi c } ##, (## m \tau \dot{a} ## is the abraham-lorentz formula for the radiation reaction).
This equation is from Griffith's book but I am a little confused by it because in relativity we have ##F_{net} = \frac{d}{dt} \gamma m v \neq m a##.
The Attempt at a Solution
First I tried to find acceleration and velocity based on time by solving the differential equation ==>
$$ a = -\frac{F}{m} e^{t/\tau} + \frac{F}{m} $$
$$\Rightarrow v= \beta + \frac{F}{m} ( t - e^{t/\tau})$$
From here I wanted to find ##t_{final}## based on d and then calculate the total energy radiated by integrating the Lienard's equation, but I am unsure since it seems like I am forgetting all I learned about relativity in my mechanics class (also the integral is hard).
So I thought I can approach from conservation of energy aspect. So total kinetic energy is ##KE_{initial} = (\gamma-1) m c^2 ##. Now part of this energy is radiated and the rest is spent on resisting the force F. So can I just write ##E_{rad} = KE_{initial} - F d ## ? That would make everything easier. Thanks in advance!