How Does Charge Density Inside a Conductor Dissolve Over Time?

[th5418]

Homework Statement

We know that free charges inside a conductor will eventually move to the conductor surface. Consider a free charge initially placed inside a conductor at t=0. Show that the free charge density $$\rho_f$$ will dissolve exponentially with time. Express the characteristic time needed to dissolve the charge in terms of the conductor's dielectric constant $$\epsilon$$ and the conductivity $$\sigma$$.

Homework Equations

I think I should use charge conservation. I'm not sure...
$$delJ + \frac{d\rho}{dt} = 0$$

The Attempt at a Solution

I know what the solution should be..
$$\rho (t) = \rho_0 e^{t}$$
where $$t=\frac{\epsilon}{\sigma}$$

 

[gabbagabbahey]

I think I should use charge conservation. I'm not sure...
$$delJ + \frac{d\rho}{dt} = 0$$

Assuming you mean [itex]\mathbf{\nabla}\cdot\textbf{J}+\frac{d\rho}{dt}=0[/tex] (i.e. div<b>J</b> not "delJ" ), that seems like a good start to me...is there some relationship between $\textbf{J}$ and $\sigma$ that might help you here?<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" /><br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I know what the solution should be..<br /> $$\rho (t) = \rho_0 e^{t}$$<br /> where $$t=\frac{\epsilon}{\sigma}$$ </div> </div> </blockquote>Surely you mean $\rho(t)=\rho_0 e^{-t/\tau}$, where $\tau\equiv\epsilon/\sigma$...right?[/itex]

 

[Reshma]

Just a hint: Rearranging the equation
$$\vec \nabla \cdot \vec J = -\frac{d \rho}{dt}$$

Can you express $$\vec J$$ in terms of $\rho (t)$ ?

 

[caduceus]

$$\vec \nabla \cdot \vec J = -\frac{\partial\rho}{\partial t}$$

and you should use the relation:

$$\vec J = \sigma\vec E$$

where $$\vec\nabla \cdot \vec E=\frac{\rho}{\epsilon}$$

 

[Count Iblis]

Ohm's law:

$$\vec{J}=\sigma \vec{E}$$

is not valid on the relevant time scale for this problem.

 

[caduceus]

No! Ohm's law is still valid. Only when the time is shorter than $$\tau$$ (which we will figure out when we solve the DE) Ohm's law turns out an invalid assumption. Because after time $$\tau$$, electrostatic equilibrium is reached, and finding $$\tau$$ is our concern.

 

[Count Iblis]

No! Ohm's law is still valid. Only when the time is shorter than $$\tau$$ (which we will figure out when we solve the DE) Ohm's law turns out an invalid assumption. Because after time $$\tau$$, electrostatic equilibrium is reached, and finding $$\tau$$ is our concern.

Ohm's law is valid on time scales much longer than the typical collision time. The time scale $$\tau$$ in this problem will be many orders of magnitude less than that.