Deduce columbs law in gauss theoram
Answers
Answer:
Explanation:Coulomb's law can be derived from Gauss' law by following the given steps:
Construct a Gaussian spherical surface of radius r and the charge q located at its centre.
Let electric field at this surface be E . ...
So, ∫E. ...
Now, using Gauss law , ∫E.
Explanation:
Gauss's Theorem: According to Gauss's theorem total number of electric lines of force passing normally through a closed surface of ray shape in an electric field(i.e., total electric flux) is equal to 1/
ε
o
times the total charge present within that surface.
i.e., Φ
E
=
ε
o
q
where, ε
o
= permitivity if free space, q in vaccum.
Derivation of Gauss's Theorem: Let +q charge is placed at a point O and a point P lies at distance r from the point O. Imagine a sphere of radius r and centre O. Thus, point P lies on the surface of the sphere. Now, the surface of the sphere will be have as Gaussian surface. Therefore, the intensity of electric field on the surface at all the points will be equal in magnitude and will be directed radially outward.
∴ The electric flux passing through the spherical surface.
Φ=E.S.cos0
o
Where S is surface area i.e., S=4πr
2
= Surface area of sphere
∴Φ
E
=E.S.
or Φ
E
=4πr
2
E .........(i)
But, by Gauss, theorem, we have
Φ
E
=
ε
o
q
.....(ii)
Hence, from equation (i) and (ii), we get
E4πr
2
=
ε
o
q
or E=
4πε
o
r
2
q
Now, imagines a charge q
o
placed at point P.
∴Force on q
o
,
F=q
o
E
F=
4πε
o
r
2
q
o
q
F=
4πε
o
1
⋅
r
2
q
o
q
.
Which is Coulomb's inverse square law.
solution