Name: Electric Energy Density
Uploaded: 2019-07-19
Duration: 4 min 11 s
Description: There is an energy associated with the electrostatic field. The expression of this electric field energy density is derived using the energy stored in a para...

Question

What is the dielectric? A dielectric slab of thickness

t

is kept between the plates of a parallel plate capacitor separated by distance

d

. Derive the expression for the capacitance of the capacitor for

t < < d

.

Accepted Answer

A dielectric is a substance which does not allow the flow of charges through it but permits them to exert electrostatics forces on another through it. A dielectric is essentially an insulator which can be polarized through the small localised displacement of its charge.

The capacitance of a parallel plate capacitor of plate area $A$ and plate separation d with vacuum between its plates if gives by

$C_{0} = \frac{ε_{0} A}{d}$

Suppose initially the charges on the capacitor plates are $\pm Q$ . Then the uniform electric field set up between the capacitor plates is

$E_{0} = \frac{σ}{ε_{0}} = \frac{Q}{A ε_{0}}$

When a dielectric slab of thickness $t < d$ is placed between the plates, the field $E_{0}$ polarises the dielectric. This induces charge $- Q_{p}$ on the upper surface and

$+ Q_{p}$ on the lower surface of the dielectric. These induced charges set up a field $E_{p}$ inside the dielectric in the opposite direction of $\vec{E_{0}}$ . The induced field is given by

$E_{p} = \frac{σ_{p}}{ε_{0}} = \frac{P}{ε_{0}}$ [ $σ_{p} = \frac{Q}{A} = P$ , polarization density]

The net field inside the dielectric is

$E = E_{0} - E_{p} = \frac{E_{0}}{k} [∵ \frac{E_{0}}{E_{0} - E_{p}} = k]$

Where $k$ is the dielectric constant of the slab. So between the capacitor plates, the field $E$ exists over a distance $t$ and field $E_{0}$ exists over the remaining distance $(d - t)$ . Hence the potential difference between the capacitor plates is

$V = E_{0} (d - t) + E t = E_{0} (d - t) + \frac{E_{0}}{k} t [∵ \frac{E_{0}}{E} = k]$

$= E_{0} (d - t + \frac{t}{k}) = \frac{Q}{ε_{0} A} (d - t + \frac{t}{k})$

The capacitance of the capacitor on introduction of dielectric slab becomes

$C = \frac{Q}{V} = \frac{ε_{0} A}{d - t + \frac{t}{k}}$

What is the dielectric? A dielectric slab of thickness $t$ is kept between the plates of a parallel plate capacitor separated by distance $d$ . Derive the expression for the capacitance of the capacitor for $t < < d$ .

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

What is the dielectric? A dielectric slab of thickness t is kept between the plates of a parallel plate capacitor separated by distance d. Derive the expression for the capacitance of the capacitor for t<<d.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

What is the dielectric? A dielectric slab of thickness $t$ is kept between the plates of a parallel plate capacitor separated by distance $d$ . Derive the expression for the capacitance of the capacitor for $t < < d$ .