21. electric charge & electric field

Fundamental Forces

• The four fundamental forces of nature are:

The gravitational interaction due to mutual attraction of all objects,

The electromagnetic interaction includes electric and magnetic forces,

The strong interaction is responsible for holding the nucleus of an atom together, and

The weak interaction is responsible for radioactivity.

Electric Charge

• Benjamin Franklin (1706-1790) suggested calling the two kinds of charge negative and positive.

• Like charges repel; Opposite charges attract.

• In a neutral atom, the number of electrons equals the number of protons in the nucleus.

The net electric charge the algebraic sum of all the charges is exactly zero.

• The number of protons or electrons in a neutral atom is called the atomic number.

• Positive ion: If one or more electrons are removed from an atom, what remains is called a positive ion.

• A negative ion is an atom that has gained one or more electrons.

• This gain or loss of electrons is called ionization.

• The principle of conservation of charge states that:  The algebraic sum of all the electric charges in any closed system is constant.

• It is a universal conservation law. No experimental evidence for any violation of this principle has even been observed.

• In any charging process, charge is not created or destroyed; it is merely transferred from one body to another.

• Quantized: Every observable amount of electric charge is always an integer multiple of this basic unit. We say that charge is quantized.

Conductors and Insulators

• Some materials permit electric charge to move easily from one region of the material to another, are called conductors, while other do not are called insulators.

• Most metals are good conductors, while most nonmetals are insulators.

• Semiconductors are intermediate in their properties.

Coulomb’s Law

• Coulomb’s law states that:  The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

• The magnitude F of the force that each of the two point charges q₁ and q₂ a distance r apart exerts on the other can be expressed as

                  F   =   k | q₁ q₂ |  /  r ²

where k is a proportionality constant.

• Point charge:  Charged bodies that are very small in comparison with the distance between them.

• The forces are repulsive, when the two charges have the same sign.

The forces are attractive, when the charges have opposite signs.

• The two forces obey Newton’s third law; they are always equal in magnitude and opposite in direction.

• The SI unit of electric charge is called one coulomb.

• The magnitude of the charge of an electron or a proton is

            e  =  1.60217656535  x  10^-19  C

• In SI units the constant k is

            k  =  8.987551787  x  10⁹  N m² / C²

• The Coulomb’s law is 

            F  =   ( 1 / 4 π ε₀ )  | q₁ q₂ |  /  r ²

where,

                        ε₀  =   8.854 x 10^-12   C² / N m²                       k  =  1 / 4 π ε₀  =   8.988 x 10⁹  N m² / C²

• Problem (E21.1):  An α particle (the nucleus of a helium atom) has mass m = 6.64 x 10^-27 kg and charge q = + 2e = 3.2 x 10^-19 C.   Compare the magnitude of the electric repulsion between two α (“alpha”) particles with that of the gravitational attraction between them.                               ( 3.1 x 10³⁵  )

• Coulomb’s law applies to any number of charges.

• The principle of superposition of forces states that: The total force acting on a charge is vector sum of the forces that each charge would exert individually.

• Coulomb’s law should be used only for point charges in a vacuum.

• Problem (E21.4):  Two equal positive charges q₁ = q₂ = 2.0 μC located at x = 0, y = 0.30 m, and x = 0, y = - 0.30 m, respectively.  What are the magnitude and direction of the total electric force that q₁ and q₂ exert on a third charge Q = 4.0 μC located at x = 0.40 m, y = 0?                                             ( 0.46  N )

Field Model

• Coulomb’s law is very useful in providing magnitude of the electric force between two point charges. When two electrically charged particles in empty space interact how does each know the other is there?

• Charge A modifies the properties of the space around it.

Charge B senses how space has been modified at its position.

The response of body B is to experience the force.

• The electric force on a charged body is exerted by the electric field created by other charged bodies.

• The field model applies to many branches of physics.

Electric Field

• The electric field at a certain point is equal to the electric force per unit charge experienced by a charge at that point.

            E  =  F₀ / q₀

The SI unit of electric field is 1 newton per coulomb ( 1 N/C ).

• The force experienced by a point charge is

            F₀  =  E q₀

• If the charge is positive, the force is in the same direction as electric field.

If the charge is negative, the force and the electric field are in opposite directions.

Electric Field due to a Point Charge

• The magnitude and direction of the electric field due to a point charge is

            E  =  ( 1 / 4 π ε₀  )   q  /  r ²  

• The electric field is inversely proportional to the inverse square distance.

Point Charge

• The field produced by a positive charge points away from the charge.

The field produced by a negative charge points towards the charge.

• The electric field strength decreases with increasing distance.

Superposition of Electric Fields

• The principle of superposition of electric fields states that:  The total electric field at a given point is the vector sum of the fields due to each point charge in a charge distribution.

Electrical Field Line

• An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric-field vector at that point.

• The field lines are directed away from positive charges and toward negative charges.

Electric Dipoles

• An electric dipole is a pair of point charges with equal magnitude and opposite sign separated by a distance.

• The electric dipole moment is directed from the negative end to the positive end of the molecule.

• The electric dipole moment is the product of the charge and the separation distance.

            p  =  q d

The units of the electric dipole moment are C m.

• The magnitude of the torque exerted by the field is

            τ   =   p E sin φ

where the angle φ is between the electric field and the dipole axis.

• The torque on a dipole in a vector form is a vector product

            τ^->   =   p^->  x E^->   

• The net force on the electric dipole in a uniform external electric field is zero.

• There is a torque that tends to rotate in the dipole in direction of the external electric field.

• The potential energy for an electric dipole in an electric field is a scalar product.

             U   =   - p^->  . E^->    =  - p E cos φ

• The potential energy has its minimum at the stable equilibrium position when the two vectors parallel.

• The potential energy has a maximum when the two vectors are antiparallel.

• Problem (E21.13):  An electric dipole in a uniform field of magnitude 5.0 x 10⁵ N/C that is directed parallel to the right.  The charges are ± 1.6 x 10^-19 C; both lie in the plane and are separated by 0.125 nm. The dipole axis is almost antiparallel and its smallest angle with the field is 35˚.  Based on the magnitude of the electric dipole moment, find the potential energy of the system.