Gauss’s Law

Gauss’s Law

• It is seen that the charge Q emanates the flux ψ which is equal to the charge Q. This is provied by Faraday's experiment. Consider a sphere of radius r and a point charge + Q located at its centre. Then the total flux radiated outwards and passing through the total surface area of the sphere is same as the charge + Q, which is enclosed by the sphere.

• Now replace the point charge by a line charge, such that the portion of the line charge enclosed by the sphere consists of same charge + Q as before. In this case too, the total flux radiating outwards remains same as Q which is the charge on the line enclosed by the sphere.

• Similarly if the point charge + Q or a part of line charge carrying + Q are moved inside the sphere anywhere, still the total flux radiating outwards from the surface of the sphere remains same as Q.

• Now instead of a sphere, any irregular closed surface is considered with total charge enclosed as + Q in any form i.e. either point, line or surface then the total flux crossing the surface of that irregular object remains same as Q, which is charge enclosed by that object.

• These observations from Faraday's experiment lead to a law called Gauss's law. From the above discussion it is clear that irrespective of the shape of the closed surface and irrespective of the type of charge distribution, the total flux passing through the closed surface is the total charge enclosed by that surface.

Statement of Gauss's law : The electric flux passing through any closed surface is equal to the total charge enclosed by that surface.

1. Mathematical Representation of Gauss's Law

• Consider any object of irregular shape as shown in the Fig. 3.6.1.

• The total charge enclosed by the irregular closed surface is Q coulombs. It may be in any form of distribution. Hence the total flux that has to pass through the closed surface is Q. Consider a small differential surface dS at point P. As the surface is irregular, the direction of   as well as its magnitude is going to change from point to point on the surface. The surface dS under consideration can be represented in the vector form in terms of its area and direction normal to the surface at the point.

Key Point : Note that the normal to the surface is in two directions but only directed outwards is considered as required. The normal going into the closed surface at point P is not required.

• The flux density at point P is   and its direction is such that it makes an angle θ with the normal direction at point P.

• The flux dψ passing through the surface dS is the product of the component of    in the direction normal to the dS and dψ.

• Mathematically this can be represented as,

• This is the flux passing through incremental surface area dS. Hence the total flux passing through the entire closed surface is to be obtained by finding the surface integration of the equation (3.6.5).

• As seen earlier,  sign indicates the integration over the closed surface and called closed surface integral. Though the integration sign is single, over the surface S it becomes double integration. Hence S is generally used along with the sign of closed surface integral.

• Such a closed surface over which the integration in the equation (3.6.6) is carried out is called Gaussian Surface.

• Now irrespective of the shape of the surface and the charge distribution, total flux passing through the surface is the total charge enclosed by the surface.

• This is the mathematical representation of Gauss's law.

• The charge enclosed may take any of the following forms :

1. If there are number of point charges Q₁,Q₂, ... ,Q_n enclosed by the surface then

Q = Q₁ + Q₂ + ... + Q_n = ∑ Q_n

Ψ = Q = ∑ Q_n ... (3.6.8)

2. If there is a line charge with line charge density ρL then,

3. If there is a surface charge with surface charge density ρS then,

4. If there is a volume charge with volume charge density ρv then,

• The common form used to represent Gauss's law mathematically is,

• If there are more thah one chaige distribution in Gaussian surface, the net charge is algebraic addition of all the individual charges.

• If there is a closed surface such that there are no charges enclosed but there are charges around the surface then net flux over the surface is zero. This is because the flux from the charges outside, passes through the surface such that the flux entering is equal to flux leaving the surface.

2. Special Gaussian Surfaces

• The surface over which is the Gauss's law is applied is called Gaussian surface. Obviously such a surface is a closed surface and it has to satisfy following conditions :

1. The surface may be irregular but should be sufficiently large so as to enclose the entire charge.

2. The surface must be closed.

3. At each point of the surface   is either normal or tangential to the surface.

4. The electric flux density D is constant over the surface at which   is normal.

3. Limitations of Gauss's Law

1. It is applicable to symmetrical problems but cannot be applied to non-symmetrical problems. The problem is said to be symmetrical if D is normal to the Gaussian surface everywhere.

2. It is applicable on Gaussian surfaces and not applicable on non-Gaussian surfaces.

3. It can be applied only if the surface encloses the volume completely.

Examples for Practice

Ex. 3.6.1 Three point charges are located in air : + 0.008 pC at (0, Ojm, + 0.005 C at (3, 0)m, and at (0, 4) m there is a a charge of - 0.009 µC. Compute total flux over a sphere of 5 m radius with centre (0, 0).

[Ans.: 0.004 µC ]

Ex. 3.6.2 Find the charge enclosed in a cube of having side of 2 m, with the edges of the cube parallel to axes x, y and z while origin is at the centre of the cube. The charge density within the cube is 50 x2 cos(π / 2 y) µC /m³

[Ans.: 84.882 µC ]

Ex. 3.6.3 Two identical line charges of PL = 25 µC/m lie on the x and y axis. Find  at point P (3, 3, 3).

Review Questions

1. State and explain Gauss's law.

2. State the conditions to be satisfied by the special Gaussian surfaces.