## Electric Potential

### Electric Potential

The electric field intensity is acting as a force on any charges it arrives upon.

Therefore in moving a unit charge from P1 to P2, work must be done against the field.

When force is applied to move an object, work is the product of the force and the distance the object travels in the direction of the force

W

P

2 

P

1

F

d l

P

2 

P

1

Q E

d l

Q P

2 

P

1

E

d l

but since the force moves the charge against th e field 

W

 

Q P

2 

P

1

E

d l

Therefore

W Q

 

P

1 

P

2

E

d l

P 1 without specifying the path P 2 E The scalar line integral of an Irrotational (conservative) E field is path-independent 

E

d l

 0

### Equipotential surfaces

A set of points with same potential forms equipotential surface. For a point charge, equipotentials are spheres at fixed radius r. Consider the plot of the electrostatic potential contours forming equipotential surfaces around the point charge superimposed over the field lines for the point charge

As we can notice the field goes into the direction of decreasing potential If the behavior of the potential is unknown, the electric intensity field can be determined by finding the maximum rate and direction of the spatial change of the potential field

E

 

V

W Q P

2

P

1  we get  

P

1 

P

2

E

d l

 

P

1 

P

2  

V

d l

P

1 

P

2 ( 

V

) 

a l dl

P

1 

P

2

dV

V

2 

V

1 Potential difference

V

21  

P

1 

P

2

E

d l

 

P

1 

P

2

Q

4  0

R

2

a R dR a R

P

2

Q

4  0

R P

1 

Q

4  0   1

R

2  1

R

1   

V P

1 

V P

2 

V

Q

4  0

R

 Absolute potential at some finite radius from a point charge fixed at the origin (reference voltage of zero at an infinite radius)

V

W Q

 Work per Coulomb required to pull a charge from infinity to the radius R

For a collection of charges of continuous distribution

V V

 

dQ

4  0

R

V V

 1 4  0  1 4  0  1 4  0

l

s

v

 

v dv

(V)

R

s ds

(V)

R

R l dl

(V)

### Review

If the electrical force moves a charge a certain distance, it does work on that charge. The change in

electric potential

over this distance is defined through the work done by this force: Work done=Charge on Q*Potential where or

potential

is shorthand for change in electric potential,

potential difference

. This is analogous to the definition of the gravitational potential energy through the work done by the force of gravity in moving a mass through a certain distance. The units of potential difference, or simply potential, are Joules / Coulomb, which are called

Volts

(V). Physically, potential difference has to do with how much work the electric field does in moving a charge from one place to another.

• Batteries, for example, are rated by the potential difference across their terminals.

In a nine difference volt battery between the the potential positive and negative terminals is precisely nine volts.

On the other hand the potential difference across the power outlet in the wall of your home is 110 volts.

## Conductors

Are caractherized by ε, μ and σ The conductivity σ (S/m or 1/Ω*m or mhos/m) -depends on the charge density ρ -depends on the temperature Ex of superconductors: yttrium-barium-copper-oxide

### Current and Current Density

• Current • Current density amount of charge (C) given time (s)  1

C

1

s J

 current(A) area(m 2 ) 

A m

2

I

 

J

d s

### Types of current

conduction currents : present in conductors and semiconductors and caused by drift motion of conduction e or holes in a media in response to an applied field ex: J= σ* E (conduction current density) displacement or electrolytic currents : is the result of migration of positive and negative ions as well known as time-varying field phenomenon that allows current to flow between plates of a capacitor.

convection currents : involve the movement of charged particles through vacuum, air nonconductive media (e in a cathode ray tube) or other

J & E V=I*R 

j V j

 

k R k I k

( V ) Conservation of charge  

J

    

t

j I j

 0 (A)

### Conduction currents

Q

I

 

Nq u

a n

s

t

Q

t

Nq u

 

s J

Nq u

  

m A

2    For most conducting materials the average drift velocity is directly proportional to el field intensity

u

  

e E

m

/

s

 

J

  

e

e E

 

E

### Conductors in static electric field

Inside a conductor ρ=0 E=0 Under static conditions the E field on a conductor surface is everywhere surface (the normal surface to of the a conductor is an equipotential surface under static conditions )

### Charactheristics of E on conductor /free space interfaces

-The tangential component of the E field on a conductor surface is zero -The normal component of the E field at a conductor /free space boundary is equal to the surface charge density on the conductor divided by the permittivity of free space Boundary Conditions at a Conductor/ /Free Space Interface

t

n

s

0

abcda E

d l

E t

w

 0 

E t

 0

s

E

d s

E n

S

  

s

S

0

or E n

 

s

 0

## Dielectrics

-Ideal dielectrics do not contain free charges -contain bound charges Induced electric dipoles The material is polarized

Polar molecules (Permanent dipole moment) Nonpolar molecules Ex: By aligning the molecules during the fabrication of a material (use E field when the material is melted and maintain it until it solidifies) we can obtain electrets

### The volume density of the electric dipole moment

P

 lim 

v

 0

n

v k

  1 

v p k

Vector sum of the induced dipole moments Polarization vector n-#of molecules per unit volume

Homogeneous & linear & isotropic media D = εE D= ε 0 E+P

### Polarization charge densities

ps

P

a n

p

  

P

-surface -volume A polarized dielectric may be replaced by an equivalent polarization surface charge density and an equivalent polarization volume charge density for field calculation

V

 1 4  0

s

 

ps ds R

 1 4  0

v

 

p dv R

Q

s

 

ps ds

v

 

p dv

 

s

P

a n ds

v

  

P dv

o