9.3 Electric field due to a plane sheet of charge

  Electrostatics

    1.0 Introduction
    2.0 Electric charge
    3.0 Coulomb's law
    4.0 Principle of superposition
    5.0 Continuous charge distribution
    6.0 Electric field
    7.0 Electric field lines
    8.0 Insulators and conductors
    9.0 Gauss's law
    10.0 Work done
    11.0 Electric potential energy
    12.0 Electric Potential
    13.0 Electric dipole

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9.3 Electric field due to a plane sheet of charge

Consider a portion of a flat thin sheet which is infinite in size with constant surface charge density $\sigma $ (charge per unit area).

For calculating the electric field at point $P$ we will draw the gaussian surface in the form of a cylinder as shown in the figure.

Since the sheet is very thin, the field has the same magnitude in opposite directions at two points equidistant from the sheet on opposite sides as shown in the figure.

As the cylinder has three surfaces. One is curved surface and the other two are plane-parallel surface. Electric field lines at the curved surface are tangential. So, the flux passing through the curved surface is zero.

At plane surface electric field has same magnitude and perpendicular to surface. Thus from gauss law we can write,
$$ES = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$
where,

$S=2S_0:$ Surface area perpendicular to the field
${q_{in}} = \sigma S_0:$ Charge enclosed within the gaussian surface

$$E\left( {2{S_0}} \right) = \frac{{\sigma {S_0}}}{{{\varepsilon _0}}}$$$$E = \frac{\sigma }{{2{\varepsilon _0}}}$$
Thus, the magnitude of the field is independent of the distance from the sheet.