ELECTRIC FIELD DUE SOME CHARGE DISTRIBUTION DERIVED BY USING GAUSS LAW.
(i) Electric field due to line charge (infinite length)
The electric field at a distance r from a line charge density
The direction is outward perpendicular to the line charge. The E a (1/r) dependence is shown in fig. 1(b)
(ii) Electric field due to cylinder
If the line charge is a cylinder of radius R, then
(a) Electric field outside
(b) Electric field on the surface
(c) Electric field inside (at a distance r from the axis)
The direction of the field is outwards (normal to the axis). The dependence of the field on r is
shown in figure.2 (b). Inside the charged cylinder,
E a r
Outside E a (1/r)
(iii) ELECTRIC FIELD DUE TO A PLANE SHEET (INFINITE DIMENSIONS)
(a) Single sheet of charge
For the surface charge density s (coulomb/metre2) the field at a distance r from the sheet is E = (s /2e0)directed towards outward normal (from a positively charged sheet) The electric field does not depend on distance.
(b) Charged metal plate
Inside charged metal plate E = 0 Outside charged metal plate E =(s/e0)
The field is normally outwards for positively charged plate and inwards for negatively charged plates.
(iv) Electric field due to a charged spherical shell
The charge Q resides on the surface of the spherical shell (radius R)
(a) Field at outside point A
(b) Field at surface point B
(c) Field at inside point C
E = 0 The variation of field with distance r from the centre O of the shell in shown in fig. The field at
the surface is maximum. And outside the shell field varies as E a 1/r2.
(v) ELECTRIC FIELD DUE TO CHARGED CONDUCTING SPHERE
The entire charge Q resides on the surface of a charged conductor. Any charge given to the interior, flows to the surface in less than a nanosecond.
So a charged conducting sphere behaves like spherical shell. Thus,
(a) Field outside E = (1/4pe0) (Q/r2
(b) Field on surface E = (1/4pe0) (Q/R2)
(c) Field inside E = 0
Special note : The surface charge density in the above case is s = Q/4pR2. In terms
of s the fields are
outside 
on surface E = (s/e0)
inside E = 0
(vi) ELECTRIC FIELD DUE TO A CHARGED SPHERE (NON CONDUCTING)
In case of a charged non conducting (plastic etc.) sphere, charges do not flow. As a result, charges exist inside the sphere as well as on the surface. Assuming uniform charge distribution, the electric field, outside (point A), on the surface (point B) and inside (point C) are as follows.
(a) Field outside
E = (1/4pe0) (Q/r2) directed radically outwards (for positive Q)
(b) Field on surface
E = (1/4pe0) (Q/R2)
(c) Field inside
E = (1/4pe0) (Q/R3) r
directed radially outwards for positive Q.
The variation of the electric field with distance r from the centre of the charged nonconducting
sphere is as shown in fig. The field outside varies inversely as square of the distance. The field
at the surface is maximum. The field inside is directly proportional to the distance.
Special note : The volume charge density in above case, is r = Q/{(4/3) pR3} In terms of r, the field are
Outside E = (r/3e0{R3 /r2}
On surface E = (r/3e0)R
Inside E = (r/3e0)r
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