General Form of Gauss’s Law
In the previous episode of this series we discussed what Gauss’s Law is and we proved the statement.
Gauss’s Law states that :
The net electric flux through any hypothetical closed surface is equal to (1/ε₀) times the net electric charge within that closed surface
So, if there is a closed surface and any point charge is put inside the closed surface ,then the net flux through the closed surface is the charge times 1/ε₀
So, the net flux φ =q/ε₀
In the previous episodes , we described flux as the number of electric field lines that cross through the surface. So, if +q charge is put inside a closed surface, the number of electric field lines that will cross the surface is φ. Now, what will happen when more than 1 point charge is put inside a closed surface? What will be the flux through the surface then? Let , two point charges +q₁ and +q₂ are put inside a closed surface .
Now, we have to calculate the total flux through the surface for these two charge. First , let us see what will happen if we put these two charges individually inside the closed surface.
If we put these two charges individually , for the charge q₁ , the number of electric lines that will cross the surface is φ1 =q1/ε₀ . And for the charge q2 , it is φ2 =q2/ε₀ . If we put both the charges inside the closed surface , then all of the electric field lines generated from both the charge q1 and q2 will cross the surface . Because no electric filed line will be lost or destroyed.
So, the total number of electric field lines that will cross the closed surface is ,
φ = φ1 + φ2
or, φ = q1/ε₀ + q2/ε₀
or, φ = (q1 + q2) / ε₀
Now, q1 + q2 = Σ q . Or the net charge.
So, φ =Σ q /ε₀
This is the general form of the Gauss’s Law. Now let us see a few more examples. If a negative charge is put inside a closed surface , then Σ q is negative. So, the net flux will be negative.
For a negative charge, all electric field lines point towards the negative charge. So, all the field lines go into the surface. In previous examples , when the charge was positive , the field lines went out of the surface, if this is considered as positive flux , then the field lines going into the surface will be considered as negative flux. So, in this case the net flux is ,
φ = - q /ε₀
If two opposite point charges but same magnitude are put inside a closed surface, then
Σ q = +q - q = 0
So, φ = 0/ε₀
or, φ = 0
As, you can see from the picture , any electric field lines that goes out of the surface , must come inside the surface . Because all electric field lines goes from positive charge to negative charge. So, the net flux will be 0.
If there is no charge inside the closed surface , but there is a point charge outside the closed surface , then the total charge inside the surface is Σ q = 0. the net flux φ = 0
As you can see from the picture, the total number of field lines that goes into the surface equals the total number of field lines that goes out of the surface. So, the net flux φ = 0.
