Matter and interactions

In this class we continue to study matter and its interaction.  The interactions we have considered so far are gravity and contact forces, such as elastic forces, normal forces, and friction and drag.  On a microscopic scale, these contact forces are electromagnetic forces.  The electromagnetic force is responsible for the atomic and molecular interactions that determine the properties of inanimate matter and living organisms.

The classical model of electromagnetic interactions works well describing many everyday phenomena.  But scientific models can change as our knowledge changes.  They are made by people for people, to help us understand observable features of of the world around us.  Starting near the beginning of the 20^th century new instruments began revealing the behavior of matter on an atomic and sub-atomic scale, and the classical model does not agree with these observations.  Quantum mechanics is the model that describes matter and its interactions on that scale.

The idea that matter is made of atoms was suggested by the Greek philosopher Democritus, who lived in the fifth century BC.  Experiments by chemists in the 19th century gave strong evidence that this indeed is the case.  In the 20th century the modern picture of the atom emerged.
Approximately ninety stable atoms make up all matter on earth.  An atom consists of a nucleus surrounded by electrons.  The nucleus is a composite particle, made up of nucleons (protons and neutrons).  The electrons are thought to be elementary particles. 

Protons, neutrons, and electrons have an intrinsic property called mass.

Nearly all the mass of an atom is concentrated in the nucleus.  The mass of a nucleon is approximately 2000 times larger than the mass of an electron.

Mass is an additive quantity.  The mass of a composite object is the sum of the masses of all its constituent.  The mass of all particles has the same sign, which is chosen to be positive.  When adding masses, the sum is always greater than the mass of the individual constituents, there is no cancellation.

Massive objects have inertia.  It takes a force to change their state of motion.  All massive object interact via the gravitation force.  A particle with mass m₁ exerts a force F₁₂ on a particle with mass m₂.  Newton's law of gravitation gives this force as

F₁₂ = (-G m₁m₂/r₁₂²) (r₁₂/r₁₂).

Here r₁₂ is the distance between particles 1 and 2, and (r₁₂/r₁₂) is the unit vector pointing from particle 1 to particle 2. 
G is the gravitational constant, G = 6.67*10^-11 Nm²/kg².

Unit vectors:  A unit vector is a vector of magnitude 1.  It has no units and is used as a direction indicator.  A unit vector pointing in the x-direction has a x-component of 1 and y- and z- components of zero. It is denoted by i or (x/x).  Similarly, a unit vector pointing in the y-direction is denoted by j or (y/y), and a unit vector pointing in the z-direction is denoted by k or (z/z).

The force F₂₁, which the particle with mass m₂ exerts on the particle with mass m₁, is equal to -F₁₂, according to Newton's third law.  The gravitational force is always attractive.

The magnitude of the gravitational force between two masses decreases as one over the square root of the distance between their centers.  We treat an object's weight near the surface of the earth as constant, because its distance from the center of the earth stays roughly constant.

Embedded Question 1

Earth and Moon attract each other via the gravitational force. Which force diagram correctly represents the magnitude and direction of the force on each of these objects?

Discuss this with your fellow students in the discussion forum!
Review Newton's laws of motion.

The Coulomb force

Protons and electrons have an another intrinsic property called charge.

Charge is also an additive quantity.  The charge of a composite object is the sum of the charges of all its constituent.  However, there are two kinds of charge, which we label positive and negative.  When adding charges, we can have cancellation.  The SI unit of charge is C = Coulomb.  The protons and electron charge are

q_proton = 1.602*10^-19 C, q_electron = -1.602*10^-19 C.

The charge of a fundamental particle may be positive or negative, but its magnitude is always an integer multiple of the fundamental quantity q_e = 1.6*10^-19 C.  Unlike mass, charge is quantized.  The electric charge of the proton is positive, the charge of the electron is negative.  Each electron has a charge of -q_e, each proton has a charge of +q_e.  Neutrons have no charge.  The net charge of a system of particles is the sum of the charges of all the particles in the system.  If we combine equal amounts of positive and negative charge we obtain zero net charge.  The net charge of a closed system is a conserved quantity.  Net charge cannot be created or destroyed.  We say that charge is conserved.

Atoms are neutral particles.  They have no net charge.  The charge of the nucleus is exactly canceled by the charge of the electrons.  We have exact cancellation, because charge is quantized.  The "atomic number" Z of an atom gives the number of protons in the nucleus.  The number of electrons in a neutral atom equals the number of protons.  The charge of the nucleus is Zq_e.  The "atomic mass" A gives the total number of nucleons.  The number of neutrons is A - Z.  Chemical and structural properties of matter are determined by the way electrons are arranged in various atoms and how atoms combine to make molecules or other structures.  The number of atoms in ordinary matter is extremely large.  For example, 18 grams of water consist of about 6*10²³ atoms of oxygen and twice that many atoms of hydrogen.  Since a neutral atom is made up of an equal number of protons and electrons, its net charge is zero.  The net charge of ordinary matter, composed of neutral atoms, is zero.  We only become aware of electrostatic interactions between macroscopic objects when the positive and negative charges on the objects do not exactly cancel, leaving a net charge on the objects.

An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge.  Ions in gases are usually produced in collisions with photons, free electrons, or other atoms and molecules.  They are usually highly reactive.  Ions also form in liquids.  Given the interaction with the molecules of the liquid, ions form if it is energetically favorable to transfer an electron from one type of atom or molecule to another.  These ions are usually quite stable.

Problem:

Estimate the total amount of positive charge in a 60 kg person.

Solution:

• Reasoning:
  The body is mostly made of water.  Each water molecule is made of two hydrogen atoms and one oxygen atom.  The nucleus of a hydrogen atom is a single proton with charge 1.602*10^-19 C.  The nucleus of an oxygen atom contains eight protons.  So each molecule has a positive charge of 10*1.602*10^-19 C.  The atomic mass of oxygen is A = 16, and the atomic mass of hydrogen is A = 1.  The mass of a water molecule therefore is 18*(1.7*10^-27 kg) = 3*10^-26 kg, where 1.7*10^-27 kg is the mass of a nucleon.  The number of molecules in the person is N = 60 kg/(3*10^-26 kg) = 2*10²⁷.  The positive charge in the person is 2*10²⁷*10*1.6*10^-19 C = 3.2*10⁹ C.

All charged particles interact via the Coulomb force.  A particle with charge q₁ exerts a force F₁₂ on a particle with charge q₂. Coulomb's law gives this force as

F₁₂ = (k_eq₁q₂/r₁₂²) (r₁₂/r₁₂).

The constant k_e is k_e = 9*10⁹ Nm²/C².  It is often written as k_e = 1/(4πε₀), where ε₀ = 8.85*10^-12 C²/(Nm²) is called the permittivity of free space.
The force F₂₁, which the particle with charge q₂ exerts on the particle with charge q₁, is equal to -F₁₂, according to Newton's third law.  Two positively charged particles repel each other.  Two negatively charged particles repel each other.  But a positively charged particle and a negatively charged particle attract each other.

Problem:

What is the force between two objects, each with mass 1 kg and charge +1 C, positioned on the x-axis at x = -0.5 m and x = 0.5 m, respectively?

Solution:

• Reasoning:
  The two objects interact via the gravitational and the Coulomb force.  The net force is the vector sum of those two forces.
• Details of the calculation:
  The gravitational force on the object at x = 0.5 m is F₁₂ = -(Gm₁m₂/r₁₂²) (r₁₂/r₁₂) = -G(1 kg²)/(1 m²)i = -6.67*10^-11 N i.  (Newton's law of gravitation)
  The force is attractive.  The force on the object at x = -0.5 m has the same magnitude, but opposite direction.  The gravitational force pulls the two masses towards each other.
  
  The Coulomb force on the object at x = 0.5 m is F₁₂ = (k_eq₁q₂/r₁₂²) (r₁₂/r₁₂) = k_e(1 C²)/(1 m²)i = 9*10⁹ N i.  (Coulomb's law)
  The force is repulsive. The force on the object at x = -0.5 m has the same magnitude, but opposite direction.  The Coulomb force pushes the two masses away from each other.
  
  The magnitude of the Coulomb force in this example is so much bigger than the magnitude of the gravitational force that the gravitational force can safely be neglected.

Problem:

Compare the strengths of the electric and the gravitational force between a proton and an electron.

Solution:

• Reasoning:
  For the gravitational force we have
  F₁₂ = (-G m₁m₂/r₁₂²) (r₁₂/r₁₂),
  and for the electric force we have
  F₁₂ = (k_eq₁q₂/r₁₂²) (r₁₂/r₁₂).
• Details of the calculation:
  The ratio of F_e/F_g therefore is given by
  F_e/F_g = k_eq₁q₂/(Gm₁m₂)
  = 9*10⁹*(1.6*10^-19)²/(6.67*10^-11*1.67*10^-27*9.1*10^-31) = 2.27*10³⁹.

When considering the interactions of atomic constituents, gravity is entirely negligible.  It becomes only important on a very large scale.  This is because the masses of all the constituents of ordinary matter add.  Because we live on the surface of a very massive object,we are very aware of the gravitational force.  We usually do not observe the electric force between macroscopic objects because ordinary matter is overall electrically neutral.  But on a microscopic scale gravity can be neglected and the electric force dominates.

The total force on an object is the vector sum of all the forces acting on it.  The total electric force is the vector sum of all the electric forces acting on the object.  This is called the principle of superposition.  To review how to add vectors, explore this link.

External link: Vectors - Fundamentals and Operations

If we have 4 charged particles, with charges q₁, q₂, q₃, and q₄, respectively, then the total electric force acting on particle 1 is
F₁ = F₂₁ + F₃₁ + F₄₁
= -(k_eq₁q₂/r₁₂²) (r₁₂/r₁₂) - (k_eq₁q₃/r₁₃²) (r₁₃/r₁₃) - (k_eq₁q₂/r₁₄²) (r₁₄/r₁₄).

Problem:

Consider 3 positive charges q at the vertices of an equilateral triangle.  Each side has length l.  Find the total force on each of the charges.

Solution:

• Reasoning:
  The force on q₁ is F₁ = F₂₁ + F₃₁ (vector sum).
• Details of the calculation:
  The horizontal components of F₂₁ and F₃₁ cancel and the vertical components add.  F₁ points upward, in the y-direction.  The magnitude of F₁ is F₁ = F₂₁cos30^o + F₃₁cos30^o.
  The magnitudes F₂₁ and F₃₁ are given by F₂₁ = F₃₁ = k_eq²/l².
  F₁ = 2 k_e(q²/l²)cos30^o in the y-direction, F₁ = 2 k_e(q²/l²)cos30^o j.
  F₂ and F₃ have the same magnitude.  If F₁ points north, then F₂ points 30^o south of west and F₃ points 30^o south of east.

Triboelectricity

Charge cannot be created or destroyed, but charge can be separated.  In atoms, the outermost electrons are bound to the nucleus by the weakest net forces.  They have the largest distance from the nucleus, and they are also repelled by the inner electrons.  Some atoms attract their outermost electrons more strongly than other atoms.  For example, atoms in rubber attract their outermost electrons more strongly than atoms in wool.  When a rubber rod is rubbed with wool, electrons near the surface are attracted more strongly to the rubber than to the wool, and some electrons are transferred from the wool to the rubber.  The rubber becomes negatively charged and the wool becomes positively charged.  When enough electrons have been transferred, then the wool becomes more attractive and the rubber more repulsive to additional electrons, and the transfer stops.  This phenomenon is called triboelectricity or contact electricity.

The triboelectric sequence classifies materials according to the ease with which they become electron donors or acceptors. 

Please explore this simulation:  Balloons and Static Electricity
You can use the Balloon simulation to show repulsion, but it is a little tricky.  You have to lightly charge one balloon and then quickly charge the other one.  Then the second one repels the first one and vice versa.  Make sure the balloon does not touch the wall.  You can grab it and place it somewhere else.

Every force that we experience in everyday life, except for the gravitational pull of the earth, is electromagnetic in nature.  Electromagnetic interactions govern chemistry, determine the strength of materials, and produce light.  Electrostatics, describing the interaction between static charged objects, is a part of classical electromagnetic theory.  Electrostatics is the basis of several industrial processes.  The unintentional separation of positive and negative charges can lead to different types of industrial accidents by producing sparks.  Electrostatics forces are also important in many life processes.