Physics > Electrostatics > 9.0 Gauss's law
Electrostatics
1.0 Introduction
2.0 Electric charge
3.0 Coulomb's law
3.1 Coulomb's law in vector relations
3.2 Comparision between coulomb's force and gravitational force
4.0 Principle of superposition
5.0 Continuous charge distribution
6.0 Electric field
6.1 Electric field due to a point charge
6.2 Electric field due to a ring of charge
6.3 Electric field due to a line of charge
7.0 Electric field lines
8.0 Insulators and conductors
9.0 Gauss's law
9.1 Electric field due to a point charge
9.2 Electric field due to a linear charge distribution
9.3 Electric field due to a plane sheet of charge
9.4 Electric field near a charged conducting surface
9.5 Electric field due to a charged spherical shell or solid conducting surface
9.6 Electric field due to a solid sphere of charge
10.0 Work done
10.1 Work done by electrical force
10.2 Work done by external force
10.3 Relation between work done by electrical & external force
11.0 Electric potential energy
12.0 Electric Potential
12.1 Properties
12.2 Use of Potential
12.3 Potential Due to Point Charge
12.4 Potential due to a Ring
12.5 Potential Due to Uniformly charged Disc
12.6 Potential Due To Uniformly Charged Spherical Shell
12.7 Potential Due to Uniformly Charged Solid Sphere
13.0 Electric dipole
13.1 Electric field due to a dipole at axial point
13.2 Electric field on equatorial line
13.3 Electric field at any point
13.4 Dipole in an external electric field
13.5 Potential due to an electric dipole
9.5 Electric field due to a charged spherical shell or solid conducting surface
3.2 Comparision between coulomb's force and gravitational force
6.2 Electric field due to a ring of charge
6.3 Electric field due to a line of charge
9.2 Electric field due to a linear charge distribution
9.3 Electric field due to a plane sheet of charge
9.4 Electric field near a charged conducting surface
9.5 Electric field due to a charged spherical shell or solid conducting surface
9.6 Electric field due to a solid sphere of charge
10.2 Work done by external force
10.3 Relation between work done by electrical & external force
12.2 Use of Potential
12.3 Potential Due to Point Charge
12.4 Potential due to a Ring
12.5 Potential Due to Uniformly charged Disc
12.6 Potential Due To Uniformly Charged Spherical Shell
12.7 Potential Due to Uniformly Charged Solid Sphere
13.2 Electric field on equatorial line
13.3 Electric field at any point
13.4 Dipole in an external electric field
13.5 Potential due to an electric dipole
The electric field at any point $P$ which is at a distance $r$ from the spherical shell of radius $R$ and charge $q$ which is uniformly distributed over the surface is given as follows,
$$\begin{equation} \begin{aligned} {E_{{\text{inside}}}} = 0 \\ {E_{{\text{surface}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\frac{q}{{{R^2}}} \\ {E_{{\text{outside}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\frac{q}{{{r^2}}} \\\end{aligned} \end{equation} $$
Proof: Electric field inside the spherical shell
At all points inside the charged spherical conductor or hollow spherical shell, electric field is zero $\left( {\overrightarrow E = 0} \right)$ as there is no charge inside the conducting shell. From the property of the conductor we know that the interior of a conductor has no excess charge. Therefore,
$$E = 0$$
Proof: Electric field at any external point outside the spherical shell
Consider a spherical shell of radius $R$ and having charge $q$ which is uniformly distributed over its surface. For calculating the electric field at any external point $P$ which is at a distance $r$ from the centre of the shell, we will draw a gaussian surface in the shape of a sphere as shown in the figure.
At all point, the electric field is perpendicular to the surface. So, from the Gauss's law, we can write,
$$ES = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$
where,
$$ES = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$
where,
$S = 4\pi {r^2}:$ Surface area of the gaussian surface
${q_{in}} = q:$ Charge enclosed within the gaussian surface
${q_{in}} = q:$ Charge enclosed within the gaussian surface
$$E\left( {4\pi {r^2}} \right) = \frac{q}{{{\varepsilon _0}}}$$$$E = \frac{1}{{4\pi {\varepsilon _0}}}\frac{q}{{{r^2}}}$$
Hence, the electric field at any external point is same as if the total charge is concentrated at the centre.
At the surface of sphere $r=R$, $$E = \frac{1}{{4\pi {\varepsilon _0}}}\frac{q}{{{R^2}}}$$
The variation of electric field $(E)$ with the distance from the centre $(r)$ is as shown in the figure,
