Gauss Law of Electrostatics class 12

Introduction to Gauss's Law

    Gauss's Law plays a crucial role in Physics. It is a key principle in electromagnetism.

    Gauss's Law shows the relation between electric flux through a closed surface and the charge enclosed by that surface.

Definition of Gauss's Law

Gauss's Law states that the electric flux through a closed surface is proportional to the total charge enclosed by that surface.

`\phi \prop q`

Formula

`\phi = \frac {q}{\epsilon_0}`

Where,

`\phi =` electric flux

    q = electric charge

`\epsilon_0 =` vacuum permittivity

Carl Friedrich Gauss Biography

    Carl Friedrich Gauss (1777 - 1855) is a German mathematician, astronomer, and physicist. He made significant contributions to algebra, number theory, statistics, geometry, and physics.

Full Name - Hohann Friedrich Carl Gauss

Born - 30 April 1777 

Place of Birth - Braunschweig, Germany

Died - 23 February 1855 (77 years old)

Place of death - Gottingen, Hanover Germany

Importance of Gauss's Law in Electrostatics

`\star`        Uses of Gauss's Law in Electrostatics are given below -

`\star`        It law is the foundation for electrostatic analysis in physics.

`\star`        It is used to understand the electric flux and electric fields.

`\star`        Gauss's Law relates the electric field to an enclosed charge.

`\star`        It simplifies calculations of electric fields for symmetrical charge distributions in electrostatics.

Importance of Gauss's Law in Electrostatics

    -    Uses of Gauss's Law in Electrostatics are given below -

    -    It law is the foundation for electrostatic analysis in physics.

    -    It is used to understand the electric flux and electric fields.

    -    Gauss's Law relates the electric field to an enclosed charge.

    -    It simplifies calculations of electric fields for symmetrical charge distributions in electrostatics.

Fundamentals of Electrostatics

Electric Charge and Its Properties

    Electric charge is a fundamental property of any matter that causes it to experience a force of attraction or repulsion when placed in an electromagnetic field.

There are two types of electric charges -

        Positive Charge and

        Negative Charge

Important points

        Like charges repel each other

        Opposite charges attract each other

Four Properties of Electric Charge

Additivity of charges - Charges can be added algebraically

Quantization of charges - Charges exist in discrete quantities

Conservation of charges - The total charge in an isolated system remains constant.

Interaction - Charges exert forces on one another in an electromagnetic field.

Applications of Gauss's Law 

Gauss's Law is used for calculating -

(1)    Electric field intensity due to a uniformly charged conducting sphere.
(2)    Electric field intensity due to a uniformly charged non-conducting sphere.
(3)    Electric field intensity due to infinite line charge.
(4)    Electric field due to an infinite uniformly charged non-conducting sheet.
(5)    Electric field due to an infinite uniformly charged conducting sheet.

Gauss's law

    The total electric flux passing through a closed surface kept in an electric field in vacuum or air is equal to the product of the net charge `\Sigma q` inside the volume of the closed surface and `\frac{1}{\epsilon_0}`

Thus, the total flux

`\phi = \frac{\Sigma q}{\epsilon_0}`        ......eq.(1)

Here 

`\Sigma q ` is the algebraic sum of charges that exist inside the surface.

`\epsilon_0` is a permittivity of vacuum

We know that

`\frac{1}{4 \pi \epsilon_0} = K`

`\frac{1}{ \epsilon_0} = 4 \pi K`

then from Equation 1

`\phi = \frac{1}{\epsilon_0}\times \Sigma q`

`\phi = 4 \pi K \times \Sigma q`

    The total electric flux passing through a closed surface kept in an electric field in vacuum or air is equal to the product of the net charge `\Sigma q` inside the volume of the closed surface and `\phi = 4 \pi K`.

Important Points

`\star`    Gauss's law is valid only for those regions that follow Coulomb's inverse square law.

`\star`    Gauss's law is followed in both vacuum and medium.

`\star`    The value of outgoing flux from a closed surface does not depend on the size and shape both of closed surfaces.

`\star`    The value of flux `\phi` does not depend on the position of charge in the closed surface.

`\star`    The value of flux `\phi` does not depend on the distribution of charge inside of charge.

`\star`    The value of flux `\phi` depends on the quantity and nature of the charge as well as the medium also.

Electrostatics Class 12 Chapter 1 Detailed Notes

MCQ on Gauss Law

1. The total electric flux passing through a closed surface in an electric field is equal to:

   A) The product of net charge and permittivity

   B) The product of net charge and magnetic field strength

   C) The sum of net charge and permittivity

   D) The sum of net charge and magnetic field strength

Answer: A) The product of net charge and permittivity

2. Gauss's law is valid for regions that follow:

   A) Ampere's law

   B) Coulomb's inverse square law

   C) Faraday's law

   D) Ohm's law

Answer: B) Coulomb's inverse square law

3. Gauss's law is applicable in:

   A) Vacuum only

   B) Medium only

   C) Both vacuum and medium

   D) Neither vacuum nor medium

Answer: C) Both vacuum and medium

4. The value of outgoing flux from a closed surface depends on:

   A) Size and shape of the surface

   B) Position of charge inside the surface

   C) Distribution of charge inside the surface

   D) None of the above

Answer: D) None of the above

5. The value of flux does not depend on the:

   A) Position of charge in the closed surface

   B) Quantity and nature of the charge

   C) Permittivity of the medium

   D) Size of the closed surface

Answer: D) Size of the closed surface

6. Which equation represents Gauss's law?

   A) `\phi = \frac{1}{\epsilon_0}\times \Sigma q`

   B) `\phi = \frac{\Sigma q}{\epsilon_0}`

   C) `\phi = 4 \pi K \times \Sigma q`

   D) All of the above

Answer:    D) All of the above

7. What is the permittivity of vacuum denoted by?

   A) `\Sigma q`

   B) `\epsilon_0`

   C) K

   D) `\phi`

Answer: B) `\epsilon_0`

8. The value of flux depends on the:

   A) Distribution of charge inside the closed surface

   B) Distance between charges inside the closed surface

   C) Total charge enclosed by the closed surface

   D) Electric field strength inside the closed surface

Answer: C) Total charge enclosed by the closed surface

9. The value of \phi in Gauss's law is equal to:

   A) `1/\epsilon_0`

   B) `\frac{\Sigma q}{4 \pi K}`

   C) `\frac{4 \pi K}{\Sigma q}`

   D) `4 \pi K`

Answer: D) `4 \pi K`

10. Gauss's law relates electric flux to the:

    A) Magnetic field

    B) Electric field

    C) Electric potential

    D) Electric current

Answer: B) Electric field

Related Questions for Practice 

1. What is the equation for the total electric flux passing through a closed surface in an electric field in vacuum or air? (ϕ = (Σq) / ε₀)

2. What is the permittivity of vacuum? (ε₀)

3. Rewrite the equation for total flux (ϕ) in terms of permittivity of vacuum (ε₀). (ϕ = 4πK `\times` Σq)

4. What does the value of outgoing flux from a closed surface depend on? (It does not depend on the size and shape of the closed surface.)

5. Does Gauss's law hold true in both vacuum and medium? (Yes)

6. What law does Gauss's law rely on? (Coulomb's inverse square law)

7. Does the value of flux (ϕ) depend on the position of charge within the closed surface? (No)

8. Does the value of flux (ϕ) depend on the distribution of charge inside the charge? (No)

9. What does the value of flux (ϕ) depend on? (The quantity and nature of the charge, as well as the medium)

10. Rewrite the equation for the permittivity of vacuum (ε₀) in terms of the constant K. (1 / ε₀ = 4πK)

11. What is the equation for the total electric flux passing through a closed surface in an electric field in vacuum? (Answer: `\phi = \frac{\Sigma q}{\epsilon_0}`)

12. If the net charge inside a closed surface is 5 C and the permittivity of vacuum is `8.854 \times 10^-12 C^2/(N·m^2)`, what is the total flux passing through the surface? (Answer: `{5 C} / {8.854 \times 10^-12} \frac{C^2}{N·m^2}`)

13. If the value of K (Coulomb's constant) is `9 \times 10^9 \frac{Nm^2}{C^2}` and the net charge inside a closed surface is 8 C, what is the total flux passing through the surface? (Answer: 8 C `\times` 4πK)

14. True or False: Gauss's law is valid only for regions that follow Coulomb's inverse square law. (Answer: True)

15. True or False: Gauss's law is followed in both vacuum and medium. (Answer: True)

16. Does the value of outgoing flux from a closed surface depend on the size and shape of the surface? (Answer: No)

17. Does the value of flux (`\phi`) depend on the position of the charge inside a closed surface? (Answer: No)

18. Does the value of flux (`\phi`) depend on the distribution of charge inside the closed surface? (Answer: No)

19. Does the value of flux (`\phi`) depend on the quantity and nature of the charge? (Answer: Yes)

20. Does the value of flux (`\phi`) depend on the medium? (Answer: Yes)

21. A point charge of +4 μC is located at the center of a spherical surface with a radius of 0.5 m. What is the electric flux through the spherical surface? `(\text{Answer}: 2.88 \times 10^4  \frac{Nm^2}{C})`

22. A closed surface encloses a charge of -6 μC. If the electric flux through the surface is -9.6 x 10^3 N·m²/C, what is the surface area of the closed surface? (Answer: 1.5 m²)

23. A uniform electric field of magnitude 1.2 x 10^4 N/C passes through a cylindrical surface with an area of 0.2 m². What is the total electric flux through the surface? `(\text{Answer}: 2.4 \times 10^3  \frac{Nm^2}{C})`

24. A point charge of -10 μC is located at the center of a cube with an edge length of 0.2 m. What is the electric flux through one face of the cube? `(\text{Answer}: 1.8 \times 10^4  \frac{Nm^2}{C})`

25. An electric dipole consists of two point charges: +5 μC and -5 μC, separated by a distance of 0.3 m. What is the net electric flux through a spherical surface of radius 0.5 m centered on the dipole? (Answer: 0)

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Chapter  1:  ELECTRIC CHARGE AND FIELDS

PHYSICS NOTES

• Electric Charge, Basic Properties of Electric Charge, Conductors and Insulators, and Methods of Charging
• Continuous Charge Distribution
• Coulomb's Law, Electroscope, Properties of Charge, Quantization of Charge.
• Coulomb's Law in Vector Form
• Principle of Superposition of Charges
• Electric Field and Field Lines, Types of Electric Field, Electric Field Due to a Point Charge, Electric Filed Due to a System of Charges, Electric Field lines and their Properties
• Gauss's Law of Electrostatic, Definition of Gauss's Law, Formula, Electric Charge and its Four Properties, Applications of Garss's Law, Gauss's Law, and Important Points
• Prove of Gauss Theorem in Electrostatics
• Electric Field due to Conducting Sphere
• Electric Field due to Conducting Hollow Sphere
• Electric Field due to Non-Conducting Sphere
• Electric Field due to Infinitely Long Charged Wire
• Electric Field due to Infinite Conducting Sheet of Charge..
• Electric Field due to Non-Conducting Plane Sheet
• Electric Flux
• Definitions of Electric Dipole, Electric Field due to Electric Dipole on Axial Line..
• Electric Field Due to Electric Dipole on Equatorial Line
• Torque on a Dipole in Uniform Electric Field
• Work Done in Rotating a Dipole..
• Dielectric Constant