Permittivity in Gauss' law and Ampere's law

[Conductivity]

In the derivation of the electric field inside a non conducting sphere, We still use the permittivity of free space even though we are in a medium.

The same applies for ampere's law in a solid wire.

http://physics.bu.edu/~duffy/semester2/c15_inside.html

https://stuff.mit.edu/afs/athena/course/8/8.02-esg/Spring03/www/8.02ch24we.pdf

Why is that?

 

[Charles Link]

The physics of the electric field $E$ and magnetic field $B$ is unchanged by the presence of charged particles and currents. $\\$ Electric fields are generated by both free electric charges and polarization charges, where polarization charge density is $\rho_p=-\nabla \cdot P$. There is no way by sampling the electric field and telling which type of electric charge it originated from. $\\$ Similarly with the magnetic field $B$ generated by electrical currents. There is no way of distinguishing whether the magnetic field $B$ originated from (free) currents in conductors, or from magnetic type currents, where magnetic current density $J_m=\nabla \times M/\mu_0$, or from polarization currents where polarization current density $J_p=\dot{P}$.

 

[ZapperZ]

In the derivation of the electric field inside a non conducting sphere, We still use the permittivity of free space even though we are in a medium.

The same applies for ampere's law in a solid wire.

http://physics.bu.edu/~duffy/semester2/c15_inside.html

https://stuff.mit.edu/afs/athena/course/8/8.02-esg/Spring03/www/8.02ch24we.pdf

Why is that?

Here, these are often simple problems with the aim of teaching you on how to use Gauss's Law. However, when you start to no longer ignore the permittivity of the medium, then you DO have to take that into account.

In a dielectric medium, Gauss's Law will take the form of

∇⋅D = q_tot

where D is now defined as the electric flux density. D is related to the electric field E via

D = εE

where ε is the permittivity of that medium. In vacuum, this relationship becomes

D = ε₀E

and you get back the Gauss's law that we know and love.

Zz.