Gaussian Surface

A Gaussian surface (also abbreviated as G.S.) is a three-dimensional closed surface that is used to determine the flux of a vector field, commonly the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface S = V (the boundary of a 3-dimensional region V) that is used in conjunction with Gauss’s law for the corresponding field (Gauss’s law, Gauss’s law for magnetism, or Gauss’s law for gravity) to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field.

What is the gaussian surface?

The Gaussian surface is an arbitrarily closed surface in three-dimensional space that is used to determine the flux of vector fields. A magnetic field, gravitational field, or electric field could be referred to as their vector field. In the examples below, an electric field is typically treated as a vector field. The gaussian surface is calculated using Gauss Law.

∮E⋅ⅆA=∮E⋅nⅆA

=∮ErrnⅆA

=∮ErdA

=Er∮dA

=Er4Πr2

The Gaussian surface is calculated using the formula above. The electric charge restricted in V is referred to as Q(V).

Let’s have a look at the Gauss Law. Gauss is a unit of magnetic induction equivalent to one-tenth of tesla in real terms. Gauss Law is also known as Gauss’s flow theorem in physics. This law is concerned with the dispersion of electric carriers, or charges that enter an electric field. As previously stated, the considered surface can be closed, constraining the volume, such as a spherical or cylindrical surface. The combination of the divergence theorem and Coulomb’s theorem is known as Gauss’s Law.

The electric field is considered in this article for clarity, as it is the most common sort of field for which the surface idea is utilised.

Gaussian surfaces are typically chosen with care to take advantage of a situation’s symmetry in order to make the surface integral calculation easier. If the Gaussian surface is chosen so that the component of the electric field along the normal vector is constant for every point on the surface, the computation will not need difficult integration since the constants that occur can be removed from the integral. It is defined as a three-dimensional closed surface on which the flux of a vector field can be determined.

Let us now take a deep dive into determining the Gaussian surface of various closed surfaces such as spheres and cylinders.

The Gaussian Surface of A Sphere

When flux or electric field is generated on the surface of a spherical Gaussian surface for a variety of reasons –

a single point of contact

On a spherical shell, uniform distribution

Charge distribution with a spherical proportion or symmetry

Consider a sphere with a radius R and a charge Q that is equally distributed. Gauss Law would be used to calculate the electric field at a distance ‘r’.

∮E.dA=qenc0

Er4πr2=Q0

Er=14π0Qr2

The Gaussian Surface of The Cylinder

A closed cylindrical surface is used to calculate the vector field or flux created by the following parameters:

• On the uniform infinite long line, there is a uniform charge.
• Charge uniformity on an infinite plate • Charge uniformity on an endlessly long cylinder

Consider a point charge P at a distance of r that contains the charge density of an infinite line charge. The line charge is the rotational axis for the cylinder of length ‘h,’ and the charge q is present inside the cylinder.

q=λh

Following that, on three different surfaces a, b, and c, the flux out of the cylindrical surface with the differential vector area dA is given as:

∅=E2πrL=λL0

For r≥R

E=2π0r

What Is A Gaussian Pillbox?

This surface is frequently used to calculate the electric field resulting from an infinite sheet of charge with uniform charge density or a slab of charge with a finite thickness. Consider a box with three components: a disc with area A at one end, a disc with equal area at the other end, and the cylinder’s side. The addition of electric flux via the aforementioned component of the surface is proportional to the enclosed charge of the pillbox, according to Gauss Law. The field near the sheet can be estimated constant; the pillbox angled so that field lines puncture the discs at the field’s ends at a straight angle.

Conclusion

Gaussian surfaces are typically chosen with care to take advantage of a situation’s symmetry in order to make the surface integral calculation easier. If the Gaussian surface is chosen so that the component of the electric field along the normal vector is constant for every point on the surface, the computation will not need difficult integration since the constants that occur can be removed from the integral. It is defined as a three-dimensional closed surface on which the flux of a vector field can be determined.