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ELECTRIC POTENTIAL
Newton’s Law of Gravitational Force
Coulomb’s Law
Both mathematically identical - Fe is a Conservative Force
Mechanics
∆U=UF-Ui=-W g
Electrostatics
∆U=UF-Ui=-W e
Electric Potential
Electric Potential
Electric Potential
E created around charge
Potential Energy per unit charge
U
V=
qo
Potential Energy
V created around charge
Electric Potential Difference - DV
∆V=Vf - Vi
∆U
∆V =
q
Units
1 volt = 1 Joule per coulomb
1 eV = 1.60x10-19 J
Using Electric Potential
∆U=q∆V
Electric Potential Energy
Relation Between Electric Potential and Electric Field
∆U=q∆V= -W
W=?
dW=F.dl
dW=qE.dl
F=qE
b r
r
r r
W = ∫ q E .d l = q ∫ E .d l
∆U − W
∆V =
=
q
q
b
a
also
a
r r
∆V = Vb − Va = − ∫ E.dl
b
a
For Uniform Electric Field
r r
∆V = Vb − Va = − ∫ E.dl
b
a
Charged Conducting Sphere
Determine the potential at a distance r
from the center of a uniformly charged
conducting sphere of a radius ro for a)
r>ro b) r=ro c) r<ro . The total charge of
the sphere is Q.
r r
∆V = − ∫ E.dl
b
a
Electric Field
Electric Potential
Electric Potential due to a Point Charge
r r
∆V = − ∫ E.dl
b
a
Electric Potential due to a Point Charge
1 Q
V=
4pe o r
Potential Due to a Group of Point Charges
Superposition Principle
VP=V1+V2+….+V6
Potential at point P?
1 Q1
1 Q2
1 Q6
VP =
+
+ ...... +
4pe O r1 4pe O r2
4pe O r6
For N charges
N
1
V = ∑ Vi =
4pe o
i =1
N
Qi
∑
i =1 ri
Potential due to a Continuous Charge Distribution
1 dq
dV =
4pe o r
Consider charge distribution is composed
of tiny differential elements of charges dq
1
V=
4pe o
V = ∫ dV
dq
∫r
1D-Object
2D-Object
dq = l dl
dq = s dA
Q
l =
L
Q
s =
A
3D-Object
dq = r dv
Q
r =
V
Potential due to a charged ring
1
dq
V=
4pe o ∫ r
dq = l dl
Q
Q
l = =
L 2pR
Potential due to a charged disk
1
dq
V=
∫
4pe o r
dq = s dA
Q
Q
s = = 2
A pR
EQUIPOTENTIAL SURFACES
• Points with the same electric potential
form “equipotential surfaces”.
• An equipotential surface must be
perpendicular to the electric field at any
point.
Electric Field From Electric Potential
r r
∆V = Vb − Va = − ∫ E.dl
b
Potential if we knew the electric field
a
r r
dV = − E.dl = − El dl
dV
El = −
dl
∂V
∂V
∂V
Ex = −
, Ey = −
, Ez = −
∂x
∂y
∂z
• The component of E-field in any direction is equal to the negative of
the rate of change of electric potential in that direction.
Electric Potential Energy of a System of Point Charges
Q: What is the electric potential energy of this
system of fixed point charges?
The electric potential energy of a system of fixed point charges is equal to
the work done to assemble the system bringing the each charge from an
infinite distance
What is the potential energy U of the system?
For N Particles
N
W = U = ∑ U ij = U12 + U13 + ...
i< j