How Do Electric Fields Differ Between Plasma and Vacuum Sheaths?

[touqra]

A plasma is described by the dielectric function

$$\epsilon (\omega) = \epsilon_0 (1-\frac{\omega_p^2}{\omega^2})$$

where $$\omega_p$$ is a constant. Any attempt to establish a voltage

$$V(t) = V cos \omega t$$ across the plasma generates a region of vacuum called the "sheath" on either side of the plasma volume.

Derive expressions for the uniform electric field $$E_p (t) = E_p cos \omega t$$ in the plasma and for $$E_s (t) = E_s cos \omega t$$ in the sheath. Assume that there is no free charge anywhere. Assume that $$\omega_p$$ is small enough that an electrostatic approximation is always valid.

I don't really understand. Isn't the electric field is stated in the question already ?

 

[Troels]

A plasma is described by the dielectric function

$$\epsilon (\omega) = \epsilon_0 (1-\frac{\omega_p^2}{\omega^2})$$

where $$\omega_p$$ is a constant. Any attempt to establish a voltage

$$V(t) = V cos \omega t$$ across the plasma generates a region of vacuum called the "sheath" on either side of the plasma volume.

Derive expressions for the uniform electric field $$E_p (t) = E_p cos \omega t$$ in the plasma and for $$E_s (t) = E_s cos \omega t$$ in the sheath. Assume that there is no free charge anywhere. Assume that $$\omega_p$$ is small enough that an electrostatic approximation is always valid.

I don't really understand. Isn't the electric field is stated in the question already ?

I would say that your task is to find the constants $$E_s$$ and $$E_p$$ in terms of $$V$$ and $$\epsilon(\omega)$$

I'd say start with the Laplace equation - but unless you have some futher specification of the geometries involved like a sketch og something, that might prove tricky

 

[touqra]

I would say that your task is to find the constants $$E_s$$ and $$E_p$$ in terms of $$V$$ and $$\epsilon(\omega)$$

I'd say start with the Laplace equation - but unless you have some futher specification of the geometries involved like a sketch og something, that might prove tricky

It's two plate of electrodes, one grounded, another at voltage V, separated by a distance
H + 2h, where h is the size of the sheath at each end of the electrode and H the size of plasma.

 

[mda]

You have two different types of dielectric, one is vacuum, the other given by the plasma equation... Given the obvious direction of the electric field, what is the relationship between Es & Ep?

Now what is the relationship between the fields and the potential?
Solve to get the absolute fields.