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The Gaussian surface is referred to as a closed surface in three-dimensional space in such a way that the flux of a vector field is calculated. These vector fields can either be the gravitational field or the electric field or the magnetic field.
- It is an arbitrary closed surface in which Gauss’s law is applied using surface integrals to calculate the total amount of a quantity enclosed within it.
- These quantities may be the amount of electric charge as the source of the electrostatic field or the amount of gravitational mass as the source of the gravitational field.
- The Gaussian surface generally helps evaluate the electric field intensity due to symmetric charge distribution.
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Key Takeaways: Gauss Flux, Gaussian Surface, Gauss Law, Gaussian pillbox, Gaussian Surface of a sphere, Gaussian Surface of a cylinder, magnetic induction
What is Gaussian Surface?
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The unit equalizing to the one-tenth of the tesla of magnetic induction is called Gauss. The Gauss Law in physics is also known as the Gauss Flux Theorem. Gauss Law describes the relationship between the net flux across a closed surface along with the amount of charge, i.e. Qenclosed in the surface enclosing volume.
- Here, the flux is defined as a percentage of the net number of lines emanating from a surface.
- Notably, Gauss Law stays constant for any closed surface, and the state of that surface is not governed by Gauss Law.
- That is, in order to calculate the flux, it is required to choose the surface to be used for the same and the chosen surface is usually referred to as a "Gaussian Surface".
- However, this surface is just a numerical apparatus; there is no real attribute that makes a surface "Gaussian"; it simply suggests that it is chosen to apply in Gauss Law. Mathematically, Gauss's law is given by:
Where Q(V) is the electric charge contained in the V.
Surface enclosing volume
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Gaussian Surface of a Sphere
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A flux or electric field is produced on the spherical Gaussian surface due to any of the following:
- A point charge
- A spherical shell with the uniform charge distribution
- Charge distribution with spherical symmetry
Let us assume a spherical shell S with uniform distribution of charge Q, radius R, and with negligible thickness. Using Gauss law we can find electric field E at a distance r from the center of the charged shell. As r < R, the net flux and the magnitude of the electric field on the Gaussian surface are zero.
The flux out of the spherical surface S with a surface area of radius r can be given as:
Gaussian Surface of a Sphere
Gaussian Surface of a Cylinder
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A flux or electric field is produced on the cylindrical Gaussian surface due to any of the following:
- Uniform distribution of charge in an infinitely long line
- Uniform distribution of charge on an infinitely long cylinder
- Uniform distribution of charge in an infinite plane
Let us consider a point charge P at a distance r having charge density λ of an infinite line charge. The axis of rotation for the cylinder of length h denotes the line charge, and the charge q enclosed in the cylinder:
\(q = λ/h \)
The flux out of the cylindrical surface with the differential vector area dA on surfaces a, b, and c are given below:
Gaussian Surface of a Cylinder
Gaussian Surface Equations
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| The Gaussian surface of a sphere | E = QA / 4 π∈0r2 |
| The Gaussian surface of a cylinder | E = λ / 2π∈0r |
What are the Steps to Determine the Electric Field by Applying Gauss Law?
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In general, the following methods are used to determine the electric field using Gauss' Law:
- Draw a diagram depicting the charge distribution.
- Determining the way in which electric field vector points take the help of the symmetry argument.
- The electric field of the point you want to know chooses the Gaussian surface that passes through the same. Choose a Gaussian surface that passes through the location where you want to know the electric field. In an ideal world, the surface should be such that the electric field is constant in magnitude and continuously creates a comparable point with the surface, making the evaluation of the flux integral straight.
- Now, before applying the Gauss Law, calculate the flux and the charge amount within the enclosed volume of the surface.
Previous Year Questions
- When a soap bubble is charged? [KCET 2020]
- The ratio of the charges given to the shells Q1:Q2:Q3 is….[JEE Main 2020]
- For a uniformly charged ring of radius RR, the electric field on its axis has the largest magnitude at... [JEE Main 2019]
- The electric field at a distance x from the axis of rotation is…. [VITEEE 2021]
- If Q is the total charge of this charge distribution, the radius R is...[VITEEE 2011]
- An Electric Dipole Has A Fixed Dipole Moment.… [JEE 2017]
Things to Remember
- When an electric field is supplied on the outer layer of the spherical Gaussian surface as a result of a uniform charge circulation within a spherical shell in a spherical symmetry.
- Whenever an electric field is generated on the outer layer of a cylindrical Gaussian surface as a result of charge transmission in a never-ending chain of uniformity in an infinite plane on an infinitely long cylinder.
- The Gaussian surface plays a vital role in Gauss law, as it follows the same. So, the angle situated in the middle of the electric field and the area vector is normally the same at every point.
- If you want to determine a Gaussian surface, then just take note of each point on the surface angle, whether the same or not.
- A Gaussian surface is referred to as a closed surface in three-dimensional space through which the flux of a vector field is calculated.
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Sample Questions
Ques: What can be the problem-solving strategy of a Gauss law? (5 Marks)
Ans: The problem-solving strategy for the Gauss law can be as followed:
- Identification of charge distribution’s spatial symmetry.
- Finding out the consequences of Gaussian surface after choosing to have the same symmetry like the charge distribution.
- Calculate the integral through the surface by assessing the flux over the Gaussian surface.
- Determine the enclosing amount of the charge within the Gaussian surface.
- Examine the charge distribution's electric field.
Ques: How many types of symmetry allow Gauss's law to deduce the electric field? (2 Marks)
Ans: There are 3 types of symmetry that are a charge distribution with spherical, cylindrical, and planer symmetry allowing the Gauss law to deduce the electric field.
Ques: Are there any surfaces that are invalid when it comes to Gaussian surfaces? (2 Marks)
Ans: Surfaces such as a square, hemisphere, and disk cannot be Gaussian since they have boundaries yet exclude three-dimensional volume. A rough Gaussian surface can be formed using infinite planes.
Ques: What is the planar symmetry of charge distribution? (2 Marks)
Ans: When charges are uniformly scattered across a large flat surface, a planar symmetry of charge density is obtained. All foci in a plane corresponding to the plane of accuse are indistinguishable from the charges in planar symmetry.
Ques: Explain in brief the basic components of the Gaussian surface. (2 Marks)
Ans: The Gaussian surface should be a closed surface in such a way that there is a clear separation between the focuses within, on, and outside the surface. Furthermore, the closed surface should be passing across areas where vector fields such as gravitational, magnetic, and electric need to be determined accurately.
Ques: Is the surface of a cube Gaussian? (2 Marks)
Ans: Among the other closed Gaussian surfaces, the cube is really the most simplified. For a vector field as an electric field, the charge is uniformly distributed across the volume of the solid shape. It is because the focus point of the cube corresponds to the focal point of a 3D charge shape, resulting in uniform charge appropriation inside the 3D square.
Ques: Is there any limitation when it comes to Gauss law? (2 Marks)
Ans: Gauss's Law is subject to a certain constraint. Gauss regulation is limited to calculating the vector field of a closed surface and not that of the electrical dipole. Indeed, an electrical dipole is nothing more than a separation of positive and negative charge. An electret, for example, is a highly durable electric dipole.
Ques: What exactly is a Gaussian Pillbox? (2 Marks)
Ans: The Gaussian pillbox, which is a surface with an infinite charge of uniform charge, is used to determine the electric field. The pillbox is a hollow cylinder with three parts: the plate on one side with area, the plate on the other end with a cylindrical side, with an equal area. Accordingly, the total electric transition via each portion corresponds to the pillbox's encased charge.
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