Energy paradox in classical electrodynamics?

[jcap]

Consider two massive charged objects at rest with a large horizontal distance $d$ between them (object $1$: mass $m_1$, charge $q_1$ and object $2$: mass $m_2$, charge $q_2$).

I apply a constant vertical force $\vec{f_1}$ upwards to object $1$ so that it gains an acceleration $\vec{a_1}=\vec{f_1}/m_1$.

The total amount of power $P_1$ that object $1$ radiates is given by the Larmor formula (see https://en.wikipedia.org/wiki/Larmor_formula):
$$P_1=\frac{2}{3}\frac{q_1^2 a_1^2}{4\pi\epsilon_0c^3}.\tag{1}$$
Now assume that object $2$ is constrained to move only in the vertical direction. If the horizontal distance $d$ between the objects is large then only the "radiative" part of the Lienard-Wiechert electric field due to object $1$ can do any work on object $2$ (see https://en.wikipedia.org/wiki/Liénard–Wiechert_potential). The vertical force $\vec{f_2}$ acting on object $2$ is given by:
$$\vec{f_2}=-\frac{q_1q_2}{4\pi\epsilon_0c^2d}\vec{a_1}.\tag{2}$$
The power received by object $2$, $P_2$, is given by:
$$P_2=\vec{f_2}\cdot\vec{v_2}.\tag{3}$$
The equation of motion of object $2$ is given by:
$$m_2 \frac{d\vec{v_2}}{dt}=\vec{f_2}.\tag{4}$$
As the vertical force $\vec{f_2}$ is constant and the object $2$ is initially at rest then integrating Eqn.(4) gives:
$$\vec{v_2}=\frac{t\vec{f_2}}{m_2}.\tag{5}$$
Substituting Eqn.(2) and Eqn.(5) into Eqn.(3) we find that the power $P_2$ received by object $2$ is given by
$$P_2=\Big(\frac{q_1q_2}{4\pi\epsilon_0c^2d}\Big)^2\frac{a_1^2t}{m_2}.\tag{6}$$
Finally, the ratio of the power received by object $2$, $P_2$, to the power emitted by object $1$, $P_1$, is given by
$$\frac{P_2}{P_1}=\frac{3}{2}\frac{q_2^2t}{4\pi\epsilon_0cd^2m_2}.\tag{7}$$
Thus eventually object $2$ receives more power than the total power emitted by object $1$.

What's gone wrong? :)

 

[hutchphd]

Without worrying about detail, you are accelerating particle 2 for a long time. One should not be surprised that non-relativistic approximations (for KE, for Larmor) eventually (large t) provide incorrect answers. No paradox

 

[Dale]

I agree with @hutchphd This is just a case of mixing relativistic and classical equations. The Lienard Wiechert fields are fully relativistic, but equation 4 is not. Eventually it will be a problem.

 

[hutchphd]

The Lienard Wiechert fields are fully relativistic

With the appropriately retarded times and distances (again maybe this is higher order ...not going there explicitly) so I believe there are multiple issues.

 

[Dale]

With the appropriately retarded times

Yes, definitely

 

[tech99]

I think need to subtract the power re-radiated by object 2. This should be half the incident power, which would explain the anomaly.

 

[hutchphd]

For long times the expression blows up like t. How does your argument follow?

 

[vanhees71]

With the appropriately retarded times and distances (again maybe this is higher order ...not going there explicitly) so I believe there are multiple issues.

Sure, but one has to be careful, for which problems the Lienard-Wiechert potentials are applicable. For sure they become problematic when the charged particle moves faster than light, which can happen when you use a non-relativistic equation of motion to calculate the particle's trajectory.

Even for correct relativistic trajectories trouble can happen, if the trajectories are not confined to a finite region in space and/or if they get asymptotically to light-like trajectories. The most (in)famous example is the case of hyperbolic motion, where the Lienard-Wiechert potentials don't work properly to provide a solution of Maxwell's equations. For details, see

https://arxiv.org/abs/1405.7729

Note that there's an erratum

https://doi.org/10.1119/1.4906577

 

[jcap]

I've put some numbers into the equations in such a way that the final velocities $v_1$, $v_2$ remain non-relativistic and only the radiative part of the Lienard-Weichert field from object $1$ to object $2$ is significant:
$$
\begin{eqnarray*}
a_1&=&1.0\times10^{10}\ \hbox{m/s^2}\\
q_1&=&1.0\ \hbox{C}\\
q_2&=&1.0\times10^{10}\ \hbox{C}\\
m_2&=&1.0\ \hbox{kg}\\
\Delta t&=&2.5\times10^{-5}\ \hbox{s}\\
d&=&7.5\times10^{6}\ \hbox{m}\\
c\Delta t/d&=&1.0\times10^{-3}\\
v_1/c&=&8.3\times10^{-4}\\
v_2/c&=&1.1\times10^{-7}\\
\end{eqnarray*}
$$
The energy $E_1$ radiated by object $1$ from $t=0$ to $t=\Delta t$ is:
$$E_1=5.5\times10^{-1}\ \hbox{Joules}$$
The energy $E_2$ received by object $2$ from $t=d/c$ to $t=d/c+\Delta t$ is:
$$E_2=5.5\times10^{2}\ \hbox{Joules}$$
Therefore object $2$ receives a thousand times more energy than was emitted by object $1$.

 

[Dale]

This calculation clearly still has an approximation which is not correct. So calculate the fully relativistic quantities and determine which of your above quantities deviates most from the correct expression.

Hint: post 4 by @hutchphd may be worth considering

 

[Delta2]

@jcap I personally would like to see more details on how you derive equation (2) in the OP.

You are basically saying that the force is constant, independent of time and independent of the locations of the particles. That's not how a particle usually interacts with an EM wave.

You make it look like a constant static E-field is accelerating the particle and that's where the root cause of your paradox is. If we take any constant force acting on a particle, then it is$$P=Fv=Fat=F\frac{F}{m}t=\frac{F^2}{m}t$$ so the power is increasing linearly with time.

 

[hutchphd]

I think you have tacitly used t and $\Delta t$ incorrectly. Please show the calculation explicitly.

 

[jcap]

I think you have tacitly used t and $\Delta t$ incorrectly. Please show the calculation explicitly.

Ok I'll be more careful with the times:-

$$E_1=\int_0^{\Delta t}P_1dt=\frac{2}{3}\frac{q_1^2a_1^2\Delta t}{4\pi\epsilon_0c^3}$$
$$E_2=\int_{d/c}^{d/c+\Delta t}P_2(t-d/c)dt$$
Change variables to $T=t-d/c$ giving
$$E_2=\int_0^{\Delta t}P_2(T)dT=\Big(\frac{q_1q_2}{4\pi\epsilon_0c^2d}\Big)^2\frac{a_1^2(\Delta t)^2}{2m_2}$$

 

[jcap]

@jcap I personally would like to see more details on how you derive equation (2) in the OP.

I'm assuming that the receiving body 2 is far away from the transmitting body 1 so that one can use the approximation that Feynman uses for EM radiation in https://www.feynmanlectures.caltech.edu/I_28.html Equation 28.6.

In terms of the notation I use here the electric field due to body $1$ at body $2$ is:
$$\vec{E_1}(t)=\frac{-q_1}{4\pi\epsilon_0c^2d}\vec{a_1}\Big(t-\frac{d}{c}\Big)$$
where $\vec{a_1}(t-d/c)$ is the vertical retarded acceleration of body $1$ and $d$ is the horizontal distance between body $1$ and body $2$.

 

[Delta2]

I'm assuming that the receiving body 2 is far away from the transmitting body 1 so that one can use the approximation that Feynman uses for EM radiation in https://www.feynmanlectures.caltech.edu/I_28.html Equation 28.6.

In terms of the notation I use here the electric field due to body $1$ at body $2$ is:
$$\vec{E_1}(t)=\frac{-q_1}{4\pi\epsilon_0c^2d}\vec{a_1}\Big(t-\frac{d}{c}\Big)$$
where $\vec{a_1}(t-d/c)$ is the vertical retarded acceleration of body $1$ and $d$ is the horizontal distance between body $1$ and body $2$.

I think that $d$ in the above formula is not the horizontal distance between the two bodies, but the total distance between the two bodies, which of course is not constant but it depends on time t as the two particles move, each with their own velocity, the distance between them changes. So this $d$ is actually $d(t)$ and it cannot be taken out of the integral, like you do in your calculation in post #13.

And if you going to argue that the distance between them remains approximately constant, no this is not the case, if the velocities are different (and both velocities linearly increasing with time since they move with constant acceleration), then the distance between them blows to infinity as time goes to infinity.

 

[jcap]

And if you going to argue that the distance between them remains approximately constant, no this is not the case, if the velocities are different (and both velocities linearly increasing with time since they move with constant acceleration), then the distance between them blows to infinity as time goes to infinity.

But if I use the figures in post #9 to calculate the transmitted energy $E_1$ and received energy $E_2$ due to an acceleration of body $1$ over interval $\Delta t$ (rather than calculating powers) then I have $c\Delta t/d \sim 10^{-3}$ so that $d$ can be taken to be an approximately constant horizontal distance.

 

[Vanadium 50]

This calculation clearly still has an approximation which is not correct.

It looks to me like there are multiple approximations. Where did equation 2 come from? Where are the magnetic fields? "d is large" - maybe it starts out large, but at large enough t, h > d. Indeed, eventually h >> d. It's not at all surprising that one of more of these approximations becomes invalid at large enough t.

It's a paradox, I tells ya!

 

[Dale]

But if I use the figures in post #9 to calculate the transmitted energy $E_1$ and received energy $E_2$ due to an acceleration of body $1$ over interval $\Delta t$ (rather than calculating powers) then I have $c\Delta t/d \sim 10^{-3}$ so that $d$ can be taken to be an approximately constant horizontal distance.

Since you are getting a paradox you know that at least one of your approximations is invalid. The correct way then is to eliminate as many of your approximations as possible. I doubt that this is the approximation causing the issue, but until you remove it you cannot be sure.

You need to eliminate your approximations, either one by one or all together, and recalculate. I would recommend eliminating them all together and then adding them back in one by one if desired.

 

[hutchphd]

Consider two massive charged objects at rest with a large horizontal distance $d$ between them (object $1$: mass $m_1$, charge $q_1$ and object $2$: mass $m_2$, charge $q_2$).

I apply a constant vertical force $\vec{f_1}$ upwards to object $1$ so that it gains an acceleration $\vec{a_1}=\vec{f_1}/m_1$.

The total amount of power $P_1$ that object $1$ radiates is given by the Larmor formula (see https://en.wikipedia.org/wiki/Larmor_formula):
$$P_1=\frac{2}{3}\frac{q_1^2 a_1^2}{4\pi\epsilon_0c^3}.\tag{1}$$
Now assume that object $2$ is constrained to move only in the vertical direction. If the horizontal distance $d$ between the objects is large then only the "radiative" part of the Lienard-Wiechert electric field due to object $1$ can do any work on object $2$ (see https://en.wikipedia.org/wiki/Liénard–Wiechert_potential). The vertical force $\vec{f_2}$ acting on object $2$ is given by:
$$\vec{f_2}=-\frac{q_1q_2}{4\pi\epsilon_0c^2d}\vec{a_1}.\tag{2}$$
The power received by object $2$, $P_2$, is given by:
$$P_2=\vec{f_2}\cdot\vec{v_2}.\tag{3}$$
The equation of motion of object $2$ is given by:
$$m_2 \frac{d\vec{v_2}}{dt}=\vec{f_2}.\tag{4}$$
As the vertical force $\vec{f_2}$ is constant and the object $2$ is initially at rest then integrating Eqn.(4) gives:
$$\vec{v_2}=\frac{t\vec{f_2}}{m_2}.\tag{5}$$
Substituting Eqn.(2) and Eqn.(5) into Eqn.(3) we find that the power $P_2$ received by object $2$ is given by
$$P_2=\Big(\frac{q_1q_2}{4\pi\epsilon_0c^2d}\Big)^2\frac{a_1^2t}{m_2}.\tag{6}$$
Finally, the ratio of the power received by object $2$, $P_2$, to the power emitted by object $1$, $P_1$, is given by
$$\frac{P_2}{P_1}=\frac{3}{2}\frac{q_2^2t}{4\pi\epsilon_0cd^2m_2}.\tag{7}$$
Thus eventually object $2$ receives more power than the total power emitted by object $1$.

What's gone wrong? :)

Please notice
$$\frac{P_2}{P_1}=\frac{3}{2}\frac{q_2^2t}{4\pi\epsilon_0cd^2m_2}.\tag{7}$$
can be rewritten
$$\frac{P_2}{P_1}=\frac {\frac{3}{2}\frac{q_2^2}{4\pi\epsilon_0d}}{m_2c^2}\frac{ct}d.\tag{8}$$

The first fraction is less than the electrostatic energy of the big mass divided by its rest mass energy. It should be tiny. You have chosen ridiculous numbers .

 

[jcap]

I accept that the electrostatic energy of body 2 must be much bigger than its rest mass energy. That is unphysical and perhaps therefore there is no paradox.