Unit 1: Electrostatics of point charges Introduction: electric - - PowerPoint PPT Presentation

• Introduction: electric charge.
• Electrostatic forces: Coulomb’s law.
• The electric field. Electric field lines.
• Electric flux. Gauss’s law.
• Work of the electric field.

Electrostatic potential energy Electric potential on a point

• Equipotential surfaces

Unit 1: Electrostatics of point charges

• Two kinds:
• positive and negative

* Tipler, chapter 21, section 21.1

• Introduction. Electric charge
• The positive charge is located in the protons and the

negative in the electrons. The quantity of any charge must be a multiple of these and means the electric charge is

• quantized. The lowest electric charge that can be isolated is

e = 1.6010-19 C. (electric charge of the proton and electron).

• In an isolated object the net electric charge is constant (law
• f conservation of charge).
• Electric charge unit: Coulomb (C)

[Q]=IT

• Neutral atom: Equal number of

protons (+) as electrons (-).

* Tipler, chapter 21, section 21.2

• Only the electrons can be removed from the atom

(+ charge remains) or be added to an atom (- charge remains).

• Introduction. Conductors and insulators

2 2 9

/ 10 , 9 4 1 C Nm k     

d

q1 q2

2 2 12

/ 10 85 , 8 Nm C



  

• Coulomb’s law quantifies electric forces between

point charges in vacuum.

• Experimental law similar to Newton’s gravity law.

In escalar form:

q1 and q2 of the same sign.

F

* Tipler, chapter 21, section 21.3

Electric forces: Coulomb’s law

F

2 2 1

4 1 d q q F   

d

q1 q2

• If the sign of charges is different, then the force is

attractive:

q1 and q2 with different sign.

F

* Tipler, chapter 21, section 21.3

Electric forces: Coulomb’s law

F

2 2 1

4 1 d q q F   

Usually, these forces must be written in vector form, according the considered referency system.

i , 1

F

q1 qi q2

i , 2

F

q3

 

 

j i , j 2 i , j j i j i , j i

u r q 4 q      F F

i , 3

F

In a system of charges, the net force

• n a charge is the vector sum of the

individual forces exerted on it by all the other charges in the system. i

F

* Tipler, chapter 21, section 21.3

Electric forces: Principle of superposition

• The electric field is a useful concept to model the

effect of an electric charge on the surrounding space.

* Tipler, chapter 21, section 21.4

The electric field

• The electric field at a point in the space is defined as

the electric force acting on the positive unit of charge placed at this point.

• The force produced by a charge q1 at the point where

a charge of 1 C is located (electric field E) is:

+

+

q1

+1 C

E=F(1 C)

The electric field

1 1 2 C

q k u r   E F

r

u

• So, if we put a charge inside an electric field, the

effect is a force acting over the charge (F=qE):

E

q(>0)

E F q 

q(<0)

E F q 

The electric field

E E E

+

[E]=M L T-3 I-1

The unit is N/C or V/m

q1

The electric field

• Electric field is a central force field. It only depends on the

charge creating the field (q1) and the distance to that point.

• E

E

• The electric field created by a system of point

charges is the vector sum of the field created by each of the charges:

 

 

i i 2 i i i i

u r q 4 1      E E

1

E

q1 q2

2

E

q3

3

E E

The electric field

E

• +

* Tipler, chapter 21, section 21.5

• The lines parallel to the field vector at each point

in the space are called “Electric field lines”.

E E

Electric field lines

+ +

• They are lines running from the positive charges (or

infinite) to the negative charges (or infinite).

Electric field lines

P

E

S E d d    S d

* Tipler, chapter 22, section 22.2

• Let us take a point P with an electric

field E. If we take a little surface (infinitesimal) dS around P, we can define the elemental electric flux through dS as (escalar quantity)

• If we consider a bigger (non infinitesimal)

surface (S), then the flux is not infinitesimal:

Electric Flux

 

 

S S

d d S E  

Nm2/C

q r

E

* Tipler, chapter 22, section 22.2

• Let’s take an spherical surface

with a point charge q at its centre.

• The electric field is pointing outside the sphere

(q>0) (inside if q<0)

Gauss’s law

• The electric field at any

point on the spherical surface (modulus) is

2

E r q k 

q r

E

S E d d    S d

* Tipler, chapter 22, section 22.2

• If we consider an infinitesimal

surface (dS) around the point

• n the surface of the sphere,

the electric flux through dS will be

• And the electric flux (Nm2/C)
• n the whole surface of the

sphere:

Gauss’s law

2 2

   q dS r q k dS r q k d d

S S S S

    

   

S E

2

4 r 

1    Q q d

volum e Enclosed i surface Closed

   

 

S E





volum e Enclosed i

q Q

• This result can be applied to any surface (not only

spheres) and is generally valid (Gauss’s law):

• The net outward flux through any closed surface

equals the net charge inside the surface divided by

0 Gauss’s law

>0 or <0 according  >0 or <0

* Tipler, capítulo 23, sección 23.3

• Gauss’s law can be applied to any closed surface,

but the calculus is easier if the surface (S) satisfies two features:

• a) The modulus of the electric field has the same

value at all points on the surface (is constant).

• b) The electric field vector has the same direction as

the surface vector at any point on the surface.

• In this way:

S S S

d E dS cos0 E dS ES      

  

E S

Using Gauss’s law to calculate E

• These two features can only be true if the problem

shows symmetrical charge distribution.

• As the Gauss (closed) surface must be “created” by

us, we will usually have to think about:

• Spherical surfaces
• Plane surfaces
• Cylindrical surfaces

Using Gauss’s law to compute E

• Work done to move a second charge q a trip will

be: (charge q over a distance dl)

• Let’s take an electric field created by a point

charge Q.

Work of the electric field

q  r P  dl  E  dr  φ  ur Q  F=qE

l d 

2 2

r 4 qQdr dr r 4 Q q dr F cos dl F l d F dW              

• If Q and q have the same sign (repulsive force) and

rA<rB (A closer to Q than B) then

Work is done spontaneously by the forces of the electric field.

• The work done by the electric force to

carry q along line L from A to B will be:

Work of the electric field

A  rB B  rA L  E Q

q

 d l



L AB

W

B A r r r r B A L AB

r qQ r qQ r qQ dr r qQ l d F W

B A B A

2

4 4 4 4          

 

 

• If Q and q have the same sign (rejecting

force) but rA>rB (B closer to Q than A) then

• work is done against the forces of the electric

field due to an external force

Work of the electric field

B  rA A  rB L  E Q

q 

d l



L AB

W

• If Q and q have opposite sign (attracting force)
• If rA< rB (A closer to Q than B) then
• If rA> rB (B closer to Q than A) then



L AB

W 

L AB

W

As general rule:

• If the work is positive, it means that the work is done

spontaneously by the forces of electric field:

Work done by the forces of electric field

• If the work is negative, it means that the work is

done against the forces of electric field:

Work done against the forces of electric field by an external force

Work of the electric field

B  rA A  rB L  E Q

q 

d l



L AB

W 

L AB

W

• WL

AB only depends on q, Q, rA and rB. So, if

we choose another line L’ going from A to B, the work done by the electric field will be the same: and and Work of the electric field. Electric potential energy



AA

W

AB L AB L AB

W W W  

'

BA AB

W W  

B A AB

U U W  

U is the electrostatic (electric) potential energy of a charge q in field due to Q

B A L AB

r qQ r qQ W 4 4    

Fields having this feature are called conservative fields or fields deriving from potential. For these fields

• Is the electrostatic (or electric) potential energy of

a charge q at a point at distance r from charge Q, which creates the field. C tells us that an infinite number of functions can be taken.

• U=0 is usually taken at r= , and then
• U represents the work done by the electric field to

move q from this point to infinite.

C r qQ U    4

r qQ U 4 

U=WA Work of the electric field. Electric potential energy

• Electric or electrostatic potential at a point is the

electrostatic potential energy that would have a charge of 1 C placed at this point: taking V=0 at infinite

• Represents the work done by the electric field to

carry 1 C from such point to infinite.

• And

circulation of E along any line from A to B

Electric potential

r Q q U V 4  







P P

l d E V  



 

B A B A

l d E V V  

Unit: Volt (V=J/C)

Equipotential surfaces

• Equipotential surfaces are those surfaces whose

points have the same electric potential.

Equipotential surfaces due to a positive and a negative electric charge

Their equation is V=k k being a constant

Equipotential surfaces (on a plane) due to a set of two charges are really 3d surfaces.

10 V 2 V 5 V 15 V

• 10 V
• 2 V
• 5 V
• 15 V

+ —

Equipotential surfaces

• As potential is constant across an equipotential

surface, the work done to move any charge q between two points (A and B) is zero: A B

) (   

B A AB

V V q W

VA=VB As The electric field must be perpendicular to equipotential surface.

Equipotential surface



  

B A B A

l d E V V  

Equipotential surfaces

Electric field lines are perpendicular to equipotential surfaces at every point in the space.