Test: Gauss's Law

Gauss law:

• According to gauss law, for any enclosed surface the flux associated is 1/ϵ_o times the total charge enclosed within the surface.
• It is mathematically expressed as : 
• The flux associated with a surface is independent of the shape and size of the surface.
• It only depends on the number of charges enclosed within that surface.
• The surface should remain closed.
• Thus for any shape, if the number of charges enclosed within the surface is the same then the flux will also be the same.

Hence, the correct option is (4)

• From Gauss law for magnetism, for a given surface the magnetic field lines entering the surface will be equal to the magnetic field lines leaving the surface. This is possible only when two poles are present.
• So, any magnetic field line entering will exit through the surface and any magnetic field exiting will enter through the surface. That means, there is no magnetic monopole where a magnetic field can start and end.
• ​Therefore, Gauss's law for magnetism concludes that magnetic monopoles do not exist.
• So, the pole of a magnet cannot be separated by breaking the magnet into two pieces.

Concept:
We know that the flux passing through a segment which makes angle θ at the center of the circle is:

Calculation:
Given if 1/4^th of the flux passes through the disk 

CONCEPT:
Gauss's Law for electric field: It states that the total electric flux emerging out of a closed surface is directly proportional to the charge enclosed by this closed surface. It is expressed as:
ϕ = q/ϵ₀
Where ϕ is the electric flux, q is the charge enclosed in the closed surface and ϵ₀ is the electric constant.
Given that:
Consider a charge 'q' placed inside a cube of side 'a'.
The electric flux according to Gauss's law,
⇒ ϕ = q/ϵ₀
If the charge enclosed is halved, then
⇒ q′ = q/2
Therefore, the new electric flux associated with this, 

Concept:

• Gauss' Law: This law gives the relation between the distribution of electric charge and the resulting electric field.
• According to this law, the total charge Q enclosed in a closed surface is proportional to the total flux ϕ  enclosed by the surface.

ϕ α Q

• The Gauss law formula is expressed by 

ϕ = Q / ϵ₀   

Here, 
Q = charge enclosed by the surface
ϵ₀ = permittivity of free space
From the above explanation, we can see that according to Gauss law
ϕ = Q/ε₀
Whereas, electric flux through small area dA is the product of the electric field and the area its passing through (dA)
∴ϕ = ∮E.dA = Q/ε₀ 
Thus the given equation represents Gauss’s Law for electricity

CONCEPT

• Electric Flux: It is defined as the number of electric field lines passing through the perpendicular unit area.
  
• Electric Flux = (Φ) = EA_⊥ [E = electric field, A = perpendicular area]
• Electric flux (Φ) = EA cos θ [where θ is the angle between area plane and electric field]
  
• The flux is maximum when the angle is 0°
• Gauss Law: According to gauss’s law, total electric flux through a closed surface enclosing a charge is 1/ϵ₀ times the magnitude of charge enclosed.

Where, Φ = electric flux, Q_in = charge enclosed the sphere, ϵ₀ = permittivity of space (8.85 × 10^-12 C²/Nm²), dS = surface area
According to gauss’s law,
The flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by ϵ₀.

Electric flux on a closed surface only depends on the enclosed charge.
∴ Option 2 is correct

CONCEPT:

• Gauss's Law for electric field: It states that the total electric flux emerging out of a closed surface is directly proportional to the charge enclosed by this closed surface. It is expressed as:

ϕ = q/ϵ₀
Where ϕ is the electric flux, q is the charge enclosed in the closed surface and ϵ0 is the permittivity of free space
The charge, q = 10 μC = 10 × 10^-6 C
From Gauss's law, electric flux  

CONCEPT:
Gauss's law:

• According to Gauss law, the total electric flux linked with a closed surface called Gaussian surface is 1/ϵ_o the charge enclosed by the closed surface.

⇒ ϕ = Q/ϵ_o 
Where ϕ = electric flux linked with a closed surface, Q = total charge enclosed in the surface, and ϵ_o = permittivity
Given Q = 1, and r = radius of the sphere
​By the Gauss law, if the total charge enclosed in a closed surface is Q, then the total electric flux associated with it will be given as,
⇒ ϕ = Q/ϵ_o     -----(1)
By equation 1 the total flux linked with the sphere is given as,

Hence, option 3 is correct.

CONCEPT:
Gauss's law:
According to Gauss law, the total electric flux linked with a closed surface called Gaussian surface is 1/ϵ_o the charge enclosed by the closed surface.
⇒ ϕ = Q/ϵ_o
Where ϕ = electric flux linked with a closed surface, Q = total charge enclosed in the surface, and ϵ_o = permittivity

Gauss's law:

• According to Gauss law, the total electric flux linked with a closed surface called Gaussian surface is 1/ϵ_o the charge enclosed by the closed surface.
• So if the total charge enclosed in a closed surface is Q, then the total electric flux associated with it will be given as,

⇒ ϕ = Qϵ_o   -----(1)

• By equation 1 it is clear that the total flux linked with the closed surface in which a certain amount of charge is placed does not depend on the shape and size of the surface. Hence, option 4 is correct.

Concept:
Gauss's law:
According to Gauss law, the total electric flux linked with a closed surface called Gaussian surface is 1/ϵ_o the charge enclosed by the closed surface.
⇒ ϕ = Q/ϵ_o
Where ϕ = electric flux linked with a closed surface, Q = total charge enclosed in the surface, and ϵ_o = permittivity

Calculation:
Given Q_A = Q_B = q, and A_A : A_B = 1 : 2
Where A_A and A_B = surface area of the cube A and the cube B respectively
By the Gauss law, if the total charge enclosed in a closed surface is Q, then the total electric flux associated with it will be given as,
⇒ ϕ = Q/ϵ_o     -----(1)
By equation 1 it is clear that the total flux linked with the closed surface in which a certain amount of charge is placed does not depend on the shape and size of the surface.
By equation 1 the total flux associated with cube A is given as,

By equation 1 the total flux associated with cube B is given as,

By equation 2 and equation 3,

Hence, option 3 is correct.