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}`
Derivation of Gauss's Law
Let a point charge q at point O is present inside a closed surface. We consider a small area dA on the surface whose midpoint is P.
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| Derivation o Gauss's Law |
`\phi = \oint_s \frac{K q}{r^2}\hat r.d\vec {A}`
`\phi = \oint_s \frac{K q}{r^2}\hat r.d\vec {A}` `(\because \hat r.d\vec A = dA cos\theta)`
`\phi = K q\oint_s \frac{dA cos\theta}{r^2}`
`\phi = K q\oint_s d\Omega`
Where `d \Omega` is a solid angle this solid angle made at point O by area dA and we know the value of the solid angle is `4 \pi`.
So we can write
`\phi = K q (4 \pi)` `(\because \oint_s d\Omega = 4 \pi)`
`\phi = \frac{1}{4 \pi \epsilon_0} q \times 4 \pi `
`\phi = \frac{1}{ \epsilon_0} q `
`\phi = \frac{q}{\epsilon_0}`
This is Gauss's law.
Electrostatics Class 12 Chapter 1 Detailed Notes
Related MCQs
1. What does the symbol Ï• represent in the equation Ï• = q/ε₀?
a) Electric field
b) Electric potential
c) Electric charge
d) Electric flux
Answer: d) Electric flux
2. In the equation Ï• = q/ε₀, what does q represent?
a) Electric field
b) Electric potential
c) Electric charge
d) Electric flux
Answer: c) Electric charge
3. What is the value of ε₀ in the equation Ï• = q/ε₀?
a) Electric field
b) Electric potential
c) Electric charge
d) Permittivity of free space
Answer: d) Permittivity of free space
4. What is the SI unit of electric charge q in the equation Ï• = q/ε₀?
a) Volts
b) Coulombs
c) Meters
d) Farads
Answer: b) Coulombs
5. Which physical property does the value of ε₀ represent?
a) Electric field strength
b) Electric potential difference
c) Electric charge density
d) Permittivity of free space
Answer: d) Permittivity of free space
6. What does Gauss's law describe?
a) The relationship between electric field and electric potential
b) The behavior of magnetic fields in a closed surface
c) The behavior of electric fields in a closed surface
d) The relationship between charge and current
Answer: c) The behavior of electric fields in a closed surface.
7. According to Gauss's law, the total electric flux passing through a closed surface is equal to:
a) The net charge inside the surface
b) The product of the net charge inside the surface and the electric constant
c) The net charge outside the surface
d) The product of the net charge outside the surface and the electric constant
Answer: b) The product of the net charge inside the surface and the electric constant
8. In Gauss's law, the electric flux passing through a closed surface is given by:
a) `\phi = \frac{1}{\epsilon_0}`
b) `\phi = \frac{\Sigma q}{\epsilon_0}`
c) `\phi = \frac{\Sigma q}{4\pi\epsilon_0}`
d) `\phi = \frac{4\pi\epsilon_0}{\Sigma q}`
Answer: b) `\phi = \frac{\Sigma q}{\epsilon_0}`
8. What is the value of the solid angle made at a point O (inside closed surface) by an area dA in Gauss's law?
a) `2\pi`
b) `3\pi`
c) `4\pi`
d) `6\pi`
Answer: c) `4\pi`
9. Which of the following quantities is related to the angle between the electric field and the area vector in Gauss's law?
a) Electric potential
b) Electric current
c) Electric flux
d) Electric resistance
Answer: c) Electric flux
10. Gauss's law is applicable to which type of surfaces?
a) Open surfaces only
b) Closed surfaces only
c) Both open and closed surfaces
d) Spherical surfaces only
Answer: b) Closed surfaces only
11. What is the relationship between the electric flux and the net charge enclosed by a closed surface?
a) Directly proportional
b) Inversely proportional
c) No relationship
d) Depends on the shape of the surface
Answer: a) Directly proportional
12. Gauss's law is a consequence of which fundamental law in electromagnetism?
a) Faraday's law of electromagnetic induction
b) Ampere's law
c) Ohm's law
d) Coulomb's law
Answer: d) Coulomb's law
13. The constant `\epsilon_0` in Gauss's law represents:
a) The speed of light
b) The permittivity of free space
c) The permeability of free space
d) The electric charge
Answer: b) The permittivity of free space
Related Questions
1. What does Gauss's law state?
2. What is the equation for the total electric flux passing through a closed surface?
3. How is Gauss's law derived?
4. What is the value of the solid angle made at point O by area dA?
5. What is the simplified equation for the electric flux according to Gauss's law?
6. What does the angle theta represent in the derivation of Gauss's law?
7. Can you explain the relationship between electric flux and the net charge inside a closed surface?
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