Question

Establish the formula for capacitance of parallel plate capacitor in presence of partially dielectric.

Answer 1

Capacity of a parallel plate condenser which is partially filled with a dielectric medium- By the distance between the plates is d and in between the plates there is a dielectric medium of thick-ness t and dielectric constant K. If each plate is given a charge $+$q then surface charge.
$\sigma =\frac{q}{A}$
Where A is the surface area of the plates. If the distance between the plates in negligible in comparison to their area then the intensity of electric field in the area filled with air between the plates.
$E_o=\frac{\sigma }{{\epsilon }_o}$
Electric field intensity within the dielectric medium $E=\frac{\sigma }{K{\epsilon }_o}$
By the definition of potential difference, the potential difference between the plates.
V$=$ Work done in moving a unit charge from one plate (negative) to another (positive) plate.
$=$ Work done in moving a unit charge a distance $(d-t)$ in air and distance t in dielectric medium
$V=E_o\times (d-t)+Et$
On substituting the values of $E_o$ and E,
$V=\frac{\sigma }{{\epsilon }_o}(d-t)+\frac{\sigma }{K{\epsilon }_o}t$
or $V=\frac{\sigma }{{\epsilon }_o}\left[\right(d-t)+\frac{t}{K}]$
or $V=\frac{q}{A{\epsilon }_o}\left[\right(d-t)+\frac{t}{K}]$
Hence capacity of the condenser
$C=\frac{q}{V}$
$C=\frac{{\epsilon }_{o}A}{\left[\right(d-t)+\frac{t}{K}]}$
This is the required expression.