A sphere radiating charges isotropically

[Joshua Benabou]

An interesting problem posed to me by a friend:

A small sphere, initially neutral, of radius $a$ emits $n$ charges $q$ of mass $m$ per unit time isotropically from its surface at a radial velocity of constant norm $v$.

Determine the spatial distribution of charges and currents at time $t$.

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Qualitative Analysis:

- The charge density and current density will be spherically symmetric.

- The magnetic field produced by a spherically symmetric current density is zero (classic)

- without loss of generality we may assume that $q>0$. Assuming the mechanism by which the charges are radiated respects the conservation of charge, the radiating sphere will develop an increasing negative charge

- any electric field generated by the charge distribution will be directed radially and will be spherically symmetric.

- At any point in time $t$, we can draw a ball of some minimal radius $R(t)>a$ enclosing all of the charges. At a distance $r>R(t)$, the E field will be zero by Gauss's law (since the charge enclosed is zero). For $a<r<R(t)$, the charge enclosed will necessarily be negative, thus the E field will be directed towards the origin, meaning charges will feel a force pulling them back towards the origin. So we might be end up with some periodic motion...

- Note that if we don't assume that the radiation mechanism conserves charge (e.g the charges are being somehow injected from the exterior into the sphere and then radiated away), we could end up with a potentially simpler problem, because enclosed charge in any Gaussian sphere of radius $r>a$ will always be positive, and thus the electric field will be directed away from origin, tending to accelerate the charges away from the origin.

- the acceleration of the charges in turn affects the charge distribution, so there will be some differential equation to solve

Quantitative Analysis:

Let $Q(r)$ be the charge contained within a sphere centered at the origin of radius $r$ (note that $Q(r)$ is time dependent).

Gauss's law gives: $E(r)=Q(r)/(4 \pi \epsilon_0r^2)$

Hence the acceleration on a charge $q$ is $a(r)=qQ(r)/(4 \pi \epsilon_0 mr^2)$.

By conservation of charge we also have that the charge density (which is spherically symmetric) is given by $j(r)=-\partial Q(r)/\partial t$. We also have that $j(r)=v(r)\rho(r)$ where $v$ is the signed speed of a charge at a distance $r$.

Finally by defintion $\rho(r)=\frac{dQ(r)}{dr}\frac{1}{4\pi r^2}$.

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In principle the above equations should be enough to solve for $Q(r)$ and thus $j(r)$. But I got some complicated partial differential equation involving second derivatives of $Q(r)$ with respect to $r$ and $t$, which didn't lead me very far. Also I'm not sure whether to assume the radiation mechanism conserves charge or not.

 

[mfb]

An interesting problem.
In general, it won't be sufficient to keep track of Q(r) and v(r), as you can have charges moving in both directions where you have to keep track of them independently. You need a density function $\rho(r,v,t)$.
I'll assume charge conservation here - Maxwell's laws require that, and I assume the problem is within these laws.

Qualitatively:
Initially, all charges will escape to infinity. Charges emitted later start at a lower potential energy, they will never catch up with charges emitted earlier. That makes the description easier because the total contained charge they see is always equal to the charge of the sphere at the time they were emitted.
It is clear that the charges can't keep escaping forever, not even as asymptotic state. At some point they reverse and hit the sphere again. This will slow down the charge build-up on the sphere. As long-term limit, I expect some fountain-like behavior.

Quantitatively:
I'll use R for the sphere radius to keep "a" for accelerations. I'll use V for the initial velocity to keep "v" for velocities at other times.
At the border between escaping and not escaping charges (t=T), we get $\displaystyle \frac 1 2 m V^2 = \frac{k n q^2 T}{R}$ where $\displaystyle k=\frac{1}{4 \pi \epsilon_0}$.
The assumption of spherical symmetry means we consider the limit n->infinity where q*n stays finite. We can replace the masses by a charge flow j=qn and a mass flow μ=nm. We can rewrite the previous equation as $\mu V^2 R = 2 k j^2 T$. T is the typical timescale of our problem.
All charges emitted before T will just see a central charge equal to $jt_0$ where $t_0$ is their emission time, and feel an acceleration of $\displaystyle -a(t)=\frac{k j^2 t_0}{\mu r(t)^2}$.
If $t_0=T$, then we get $\displaystyle -a(t)=\frac{V^2 R}{2 r(t)^2}$ and $v(T)=V$ and $r(T)=R$.

If we double V and μ and quadruple j, we make the system twice as fast but don't change the dynamics otherwise. We can rescale the system to make kT=1.
If we double R, V and j and half μ, we double the length scale but don't change the dynamics otherwise. We can rescale the system to make R=1.
Plugging this into the equations form above: $\mu V^2 = j^2$ and $-a(t)=\frac {j^2 t_0}{\mu r^2(t)}$, and combined we get $-a(t)=\frac {V^2 t_0}{r^2(t)}$ where times and velocities are now relative to the sphere and the typical time. We also know $r(t_0)=1$ and $v(t_0)=V$.
This is equivalent to the Kepler problem at zero angular momentum. There should be a solution for the border case, but no closed solution for the general case. WolframAlpha doesn't find the solution for the border case.
Charge and current density would follow from a solution and its derivatives.We can use the idea a bit longer .At $t=T+\epsilon$ the charge won't escape any more, but it will move away for a long time, and the charges will continue to follow the differential equation derived above. Based on conservation of energy, we can determine the maximal radius a charge will get: $\displaystyle \mu V^2 - 2 j^2 t = \frac{2 j^2 t }{R_{max}}$. This is equal to the semi-major axis in the Kepler problem, and we can find an analytic expression for the period, but as the objects start at finite radius it doesn't help.

The radius will start "at infinity" and quickly decrease. The smaller the radius gets the slower it decreases. At some point it will be crossed by charges emitted earlier, and after that point things get really messy.

 

[Joshua Benabou]

@mfb: interesting. so this is turning out to be more complicated than i had previously thought, which is weird since normally this problem should have a nice solution - it was an problem posed in an oral exam at the Ecole Normale Superieure (France).

The main complication is that, assuming the sphere radiates its charges away such that charge is conserved, we are going to have charges moving in different directions at the same radius, and things become quite complicated. Eventually you would have charges which would pass through the center of the radiating sphere and come out the other side, and the whole thing becomes a mess.

For this reason I wonder if I'm misinterpreting the problem.

How about we consider a much simpler situation: charges are being magically injected into the sphere of radius $a$ form the exterior (or are magically coming into existence in its interior, whatever you like), and then radiated away from its surface isotropically such that $n$ charges $q$ leave the surface per unit with a radial velocity of norm $v$.

In this case, the charges will always be moving away to infinity, and we don't get any crazyness. The are accelerated due to repulsive forces.

I tried solving the problem in this case but, as I mentioned in the OP, I ran into some mathematical nastyness in the form of an partial differential equation in two variables.

 

[mfb]

The problem is easy if you ignore the interaction between the charges.
The problem is not too hard if you keep the discussion qualitative.

Maxwell's laws done work with magic charge injection. If we ignore electromagnetism and just assume a force that scales with t/r², then WolframAlpha still doesn't find a solution.

 

[Joshua Benabou]

Hmm ok. Bottom line is this problem doesn't have a nice solution, which is annoying. Do you by chance have any interesting challenge problems in electromagnetism similar to this one?

 

[mfb]

You can study the problem in a different number of dimensions. As "infinite plate" it will have analytic solutions, at least as long as the charges don't overlap, but maybe the situation can be studied even afterwards.