I am given two charged particles of same charge at a distance of r. They initially apply force F.
Now an infinite dielectric (of dielectric constant 4) of width r2 is introduced between the particles. What will be the new force?
I find this problem confusing because I have only been told about forces when its either fully dielectric or fully vaccum given by Coulomb's law. How do we get forces when only partial space is dielectric?
Answer
A dielectric effectively behaves as if it was thicker than it is. If the dielectric constant is K and the thickness of the dielectric is t, then for calculating the force it behaves as if the thickness was t√K.
To see this let's take the example we know about where the dielectric fills the space between the charges:
In (a) the thickness of the dielectric is the same as the distance between the charges, r, so the effective thickness is r√K. If we put this in the force law we get:
F=14πϵ0Q1Q2(r√K)2=14πϵ01KQ1Q2r2
as we expect. The force is reduced by a factor of K.
Now consider (b). To get the effective distance between the charges we have to add the distance through the air, r−t, plus the effective thickness of the dielectric, t√K, so the effective distance between the charges is:
d=(r−t)+t√K
and the force is just:
F=14πϵ0Q1Q2d2=14πϵ0Q1Q2(r−t+t√K)2
This is how you get the force when the space between the charges is only partially filled by the dielectric.
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