Electrostatics Summary

XII STD PHYSICS NOTES



Chapter Two ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1 of 9




Potential energy difference  Thus the total work done to bring unit
charge from infinity to the point P is
r r 1 q
W  ∫ dW  ∫   2
dx
  40 x
q r 1 dx
W  ∫

 Electric potential energy difference 4 0 x2
between two points is the work required to  Integrating
be done by an external force in moving
charge q from one point to another.
q
W  4
0
( )
1 1

r 
q
4 r
0

 Therefore electrostatic potential is given by
Electrostatic potential energy 1 q
V
• Potential energy of charge q at a point is 40 r
the work done by the external force in
Variation of potential V with r
bringing the charge q from infinity to that
point.
ELECTROSTATIC POTENTIAL
 The electrostatic potential (V ) at any point
is the work done in bringing a unit positive
charge from infinity to that point.
W
V  , W – work done, q – charge.
q
 Also W  qV
 It is a scalar quantity.
 Unit is J/C or volt (V) POTENTIAL DUE TO AN ELECTRIC DIPOLE
POTENTIAL DUE TO A POINT CHARGE DERIVATION NOT INCLUDED




 The force acting on a unit positive charge
(+1 C) at A , is
1 q 1 1 q
F  4 x2  4 x2
0 0
The potential due to the dipole at P is the sum of
potentials due to the charges q and –q
 Thus the work done to move a unit positive
charge from A to B through a displacement
dx is
1 q
dW   dx  Using cosine law
2
40 x
 The negative sign shows that the work is
done against electrostatic force.  For r >> a

, THIS DERIVATION IS NOT INCLUDED Page 2 of 9




 Neglecting the higher order terms we get

2
r1  r
2
( 1
2a cos
r )  Thus V  V  V  V
1 2


 Similarly r2  r
2
2

( 1
2a cos
r ) n



 Thus 1 1
r  r 1
1
(
2a cos
r ) 2
1
and
Potential due to a uniformly charged spherical

shell

( )
1

1 1 2a cos 2
 For a uniformly charged spherical shell, the
r  r 1 r
electric field outside the shell is as if the
 Using the Binomial theorem and retaining entire charge is concentrated at the centre
terms up to the first order in a/r,  Thus potential at a distance r, from the shell
is
1 q
V
40 r
 Where r  R , radius of the shell
 Inside the shell, the potential is a constant
and has the same value as on its surface.
 Thus the potential is
1 q
V
40 R
Equipotential surface
 Using p=q x2a, we get  Surface with constant value of potential at
1 p cos all points on the surface.
V 2 Properties of Equipotential surface
40 r  Work done to move a charge on an
Special cases equipotential surface is zero.
 Potential at point on the axial line  Electric field is perpendicular to the surface.
At the axial point θ=0, therefore  Two equipotential surfaces never intersect.
1 p Equipotential surface of a single charge
V 2
40 r
 Potential at point on the equatorial line
At the equatorial line θ=900, thus , V=0.

POTENTIAL DUE TO A SYSTEM OF CHARGES
Equipotential surfaces for a uniform electric field
 By the superposition principle, the potential
at a point due to a system of charges is the
algebraic sum of the potentials due to the
individual charges.