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

Unit 1: Electrostatics of point charges  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

Introduction. Electric charge  Two kinds:  positive and negative  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 of conservation of charge).  Electric charge unit: Coulomb (C) [Q]=IT * Tipler, chapter 21, section 21.1

Introduction. Conductors and insulators  Neutral atom: Equal number of protons (+) as electrons (-).  Only the electrons can be removed from the atom (+ charge remains) or be added to an atom (- charge remains). * Tipler, chapter 21, section 21.2

Electric forces: Coulomb’s law  Coulomb’s law quantifies electric forces between point charges in vacuum.  Experimental law similar to Newton’s gravity law. In escalar form: q 1 and q 2 of the same sign. q 2 q 1 d F F      12 2 2 8 , 85 10 / C Nm q q 1 0   1 2 F 1 2    d 4 9 2 2 k 9 , 0 10 Nm / C   0 4 0 * Tipler, chapter 21, section 21.3

Electric forces: Coulomb’s law  If the sign of charges is different, then the force is attractive:  F F q q 1   1 2 F q 2 q 1 d 2 d 4 0 q 1 and q 2 with different sign. Usually, these forces must be written in vector form, according the considered referency system. * Tipler, chapter 21, section 21.3

Electric forces: Principle of superposition In a system of charges, the net force F on a charge is the vector sum of the i F F individual forces exerted on it by all 1 , i 3 , i the other charges in the system. q 2 F q i 2 , i    q  q    j i F F u i j , i   j , i 2 4 r 0 j , i j j q 3 q 1 * Tipler, chapter 21, section 21.3

The electric field  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 q 1 at the point where a charge of 1 C is located (electric field E) is: E=F(1 C) u r q +   +1 C 1 E F k u + q 1 1 C 2 r

The electric field  So, if we put a charge inside an electric field, the effect is a force acting over the charge (F=qE):  F q E E q(>0) q(<0)  F q E

The electric field E E E q 1 E E - +  Electric field is a central force field. It only depends on the charge creating the field (q 1 ) and the distance to that point. The unit is N/C or V/m [E]=M L T -3 I -1

The electric field  The electric field created by a system of point charges is the vector sum of the field created by each of the charges: E E    E 3  1  q   1 i E E u i   i 2 4 r 0 i i i E 2 q 2 q 3 q 1

Electric field lines  The lines parallel to the field vector at each point in the space are called “Electric field lines ”. E E E - + * Tipler, chapter 21, section 21.5

Electric field lines  They are lines running from the positive charges (or infinite) to the negative charges (or infinite). + +

Electric Flux  Let us take a point P with an electric E field E. If we take a little surface d S (infinitesimal) dS around P, we can define the elemental electric flux through dS as (escalar quantity) P Nm 2 /C    d E d S  If we consider a bigger (non infinitesimal) surface (S), then the flux is not infinitesimal:       d d E S * Tipler, chapter 22, section 22.2 S S

Gauss’s law E  Let’s take an spherical surface with a point charge q at its centre.  The electric field at any point on the spherical surface (modulus) is q q r  k E 2 r  The electric field is pointing outside the sphere (q>0) (inside if q<0) * Tipler, chapter 22, section 22.2

Gauss’s law E  If we consider an infinitesimal d S surface (dS) around the point on the surface of the sphere, the electric flux through dS will be q    E d S d r  And the electric flux (Nm 2 /C) on the whole surface of the sphere: q q q            E S d d k dS k dS  2 2 r r 0 S S S S  2 * Tipler, chapter 22, section 22.2 4 r

Gauss’s law  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 Q 1        d q E S   i Enclosed 0 0 Closed volum e surface   Q q >0 or <0 according  >0 or <0 i Enclosed volum e

Using Gauss’s law to calculate E  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:          E d S E dS cos0 E dS ES S S S * Tipler, capítulo 23, sección 23.3

Using Gauss’s law to compute 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

Work of the electric field  Let’s take an electric field created by a point charge Q.     E dl F=qE φ  q dr  P  u r r  Q  Work done to move a second charge q a trip will d l be: (charge q over a distance dl )   Q qQdr           dW F d l F dl cos F dr q dr   2 2 4 r 4 r 0 0

 Work of the electric field  d l B  The work done by the electric force to E q carry q along line L from A to B will  be: r B L   r r B B A qQ qQ qQ qQ B         L W F d l dr  AB     2 4 r 4 r 4 r 4 r A B 0 0 0 Q 0 A r r r A A A  If Q and q have the same sign (repulsive force) and r A <r B (A closer to Q than B) then  L W 0 AB Work is done spontaneously by the forces of the electric field.

Work of the electric field  A E  If Q and q have the same sign (rejecting q  force) but r A >r B (B closer to Q than A)  d l then r A L  L W 0 B AB  Q r B  work is done against the forces of the electric field due to an external force  If Q and q have opposite sign (attracting force)  If r A < r B (A closer to Q than B) then  L W 0 AB  If r A > r B (B closer to Q than A) then  L W 0 AB

 A Work of the electric field E q   d l r A L As general rule: B  Q r B  If the work is positive, it means that the work is done spontaneously by the forces of electric field:  L Work done by the forces of electric field W 0 AB  If the work is negative, it means that the work is done against the forces of electric field:  L Work done against the forces of electric field W 0 AB by an external force

Work of the electric field. qQ qQ   L W AB   4 r 4 r Electric potential energy 0 A 0 B  W L AB only depends on q, Q, r A and r B . So, if we choose another line L’ going from A to B, the work done by the electric field will be the same:   L L ' W W W AB AB AB    and and W W W 0 AB BA AA Fields having this feature are called conservative fields or fields deriving from potential.   For these fields W U U AB A B U is the electrostatic (electric) potential energy of a charge q in field due to Q

Work of the electric field. Electric potential energy qQ   U C  4 r 0  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. qQ  U=0 is usually taken at r=  , and then  U 4  r 0  U represents the work done by the electric field to move q from this point to infinite. U=W A 