Spread the love

Energy Density Of Electric Field

Energy Density Of Electric Field :- Consider a small body of mass m and charge q placed in an electric field E. When released, it accelerates in the direction of the field due to the force qE, gaining kinetic energy.

This energy comes from the electric field, indicating that the field stores energy in the space it occupies. This is known as the electric field energy. Let’s calculate the energy stored in the region where the electric field exists and find energy density of electric field.

Energy Density Of Electric Field – From where electric field energy comes ?

We know that there is an electric field around a charged object. To increase the value of charge on the body or to increase the volume of this electric field surrounding the body, we have to do work against this electric field. This work done against the electric field gets stored in the form of electric field energy (U) and the field energy per unit volume is called energy density of electric field (u).

Energy Density Of Electric Field – Derivation

For simplicity, let us consider a charged spherical shell of radius r having surface charge density σ.

Electrostatic pressure acting outward at the surface of the conductor,

$\displaystyle {{P}_{E}}=\frac{{{\sigma }^{2}}}{2{{\varepsilon }_{0}}}$ …..(1)

Outward force experienced by the spherical surface,

$\displaystyle F=PA=\frac{{{\sigma }^{2}}}{2{{\varepsilon }_{0}}}\times 4\pi {{r}^{2}}$ …..(2)

Now if we consider that the metal surface is flexible and can be compressed by applying force, then small amount of work done (dW) to compress the sphere by dr against the force given by equation (2) is :

$\displaystyle dW=Fdr=\frac{{{\sigma }^{2}}}{2{{\varepsilon }_{0}}}\times 4\pi {{r}^{2}}dr$ …..(3)

Due to compression the decrease in the volume of the sphere (or increase in the volume of electric field around the conductor) :

$\displaystyle dV=4\pi {{r}^{2}}dr$ …..(4)

From equation (3) and (4),

$\displaystyle dW=\frac{{{\sigma }^{2}}}{2{{\varepsilon }_{0}}}dV$

Total energy stored (U) in the whole electric field around the conductor,

$\displaystyle U=W=\int{dW}=\int{\frac{{{\sigma }^{2}}}{2{{\varepsilon }_{0}}}dV}$ …..(5)

Hence energy stored per unit volume of the electric field, i.e., energy density of electric field (u) :

$\displaystyle u=\frac{dW}{dV}=\frac{{{\sigma }^{2}}}{2{{\varepsilon }_{0}}}=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}$ …..(6)

Because $\displaystyle E=\frac{\sigma }{{{\varepsilon }_{0}}}$ .

Equation (6) given the required expression for energy density.

If in place of vacuum, there is some other medium, then

$\displaystyle {{u}_{m}}=\frac{{{\sigma }^{2}}}{2\varepsilon }=\frac{1}{2}\varepsilon {{E}^{2}}$

Although the derivation is done using a charged spherical shell, but it is valid for any type of charge configuration.

Energy Density Of Electric Field

Energy Density Of Electric Field – From where electric field energy comes ?

Energy Density Of Electric Field – Derivation

Like this: