Form Of Coulomb's Law When Dielectric Is Filled Partially Between The Charges

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Form of Coulomb’s Law when Dielectric is filled partially between the charges

Coulomb’s Law with partially filled Dielectric between the charges :-

Suppose two charge particles q₁ and q₂ are placed r distance apart in vacuum, then from Coulomb’s Law force acting between them:-

$\displaystyle F=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}$

Now let a dielectric medium ( of permittivity ε_r ) is partially filled between the charges up to a distance x, as shown in figure below:-

Now the question is what will be the new value of force between q₁ and q₂.

For that suppose the whole distance ‘r’ is filled with dielectric medium, then the force becomes:-

$\displaystyle F=\frac{1}{4\pi \varepsilon }\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}=\frac{1}{4\pi {{\varepsilon }_{0}}{{\varepsilon }_{r}}}\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{q}_{1}}{{q}_{2}}}{{{\left( \sqrt{{{\varepsilon }_{r}}}r \right)}^{2}}}$

Here in the presence of dielectric medium, the distance r, behaves like

$\displaystyle \sqrt{{{\varepsilon }_{r}}}r$

So in the given situation, where the dielectric is filled up to the distance x, so we treat distance x as $\displaystyle \sqrt{{{\varepsilon }_{r}}}x$ and the remaining distance as (r-x). Hence the total distance r becomes