Unit 1-3: Electric Charge, Coulomb's Law, and Electric Fields

Introduction to Electrostatics and Natural Phenomena

Lightning and thunder are significant atmospheric phenomena frequently observed in Nigeria's hot and humid environment. In 1752, Benjamin Franklin demonstrated that thunderclouds are charged with electricity. When these clouds discharge into the atmosphere, they create a massive spark known as lightning. The electric current produced during such a discharge is approximately $20\,KA$. This discharge also generates immense heat, causing temperatures to rise to about $150000^{\circ}C$ within a fraction of a second. This thermal energy is concentrated in a small area roughly $20\,cm$ in width. The rapid movement of air molecules due to this intense heat produces the sound known as thunder. The roaring sound often associated with clouds occurs when the sound of thunder is reflected by clouds, hills, or other obstacles.

Electrostatics is the study of static electric charges. It was the first branch of electricity investigated and, while once thought to have no practical value, it is now vital for numerous industrial applications. The knowledge of electrostatics is essential in designing cathode ray tubes for televisions and conducting electrical prospecting for minerals. Additionally, electrostatic principles are used in loudspeakers, microphones, and photocopying machines. A fundamental aspect of electric charges is the large force between them, known as the electrostatic or electric force. This force is responsible for holding electrons to nuclei to form atoms and for bonding groups of atoms together to form molecules, solids, and liquids.

The Nature and Properties of Electric Charge

The ancient Greeks first observed that amber, when rubbed with wool, attracted light objects like pieces of chaff, indicating it had acquired a net electric charge. William Gilbert later discovered that many other substances exhibit this effect and noted that the magnitude of the effect is roughly proportional to the area of the surface rubbed. In 1745, Du Fay discovered there are two types of electricity. Experimentation showed that two ebonite rods rubbed with fur repel each other, and two glass rods rubbed with silk also repel each other. However, an ebonite rod rubbed with fur attracts a glass rod rubbed with silk. This led to the classification of charges using positive and negative signs. By convention, glass rubbed with silk is positively charged, and ebonite rubbed with fur is negatively charged. Objects with a total charge of zero are considered electrically neutral. The fundamental rule of electrostatics is that like charges repel each other, while opposite charges attract.

Matter is composed of atoms, which consist of a positively charged nucleus surrounded by negatively charged electrons. The nucleus contains protons and neutrons; protons carry a positive charge while neutrons are neutral. Protons and electrons have equal but opposite charges. Within the nucleus, protons and neutrons are held together by a strong nuclear force that prevents protons from moving. Conversely, electrons are held by a much weaker force. Some electrons are loosely bound and can move away from the atom; these are known as free electrons. When two materials are rubbed together, electrons are transferred. For example, rubbing a plastic ruler with wool causes electrons to flow from the wool to the plastic, giving the ruler a net negative charge and the wool an equal positive charge due to the electron deficit.

Units, Conservation, and Quantization of Charge

In the International System of Units (SI), the unit of electric charge is the coulomb ($C$). The charge of a single proton is $+1.6 \times 10^{-19}\,C$, and the charge of an electron is $-1.6 \times 10^{-19}\,C$. The notation for the amount of charge is $q$. A coulomb is defined as the quantity of charge flowing per second through a conductor with a steady current of $1\,A$. This relationship is expressed by the formula:

$q = It$

Where $q$ is charge in coulombs, $I$ is current in amperes, and $t$ is time in seconds. Another definition of the coulomb is the amount of charge that produces a force of $9.0 \times 10^9\,N$ between two point-like objects separated by $1\,m$.

Electric charge is a conserved quantity. The law of conservation of charge states that the total electric charge in an isolated system—the algebraic sum of positive and negative charges—does not change. During charging by friction, no new charges are created; rather, they are transferred, ensuring the net charge remains constant. Furthermore, charge is quantized, meaning it exists in discrete packets rather than continuous amounts. The fundamental unit of charge is the magnitude of the charge of an electron or proton, denoted as $e$. Any detectable charge $q$ is an integral multiple of this fundamental charge, expressed as:

$q = ne$

In this equation, $n$ is a positive or negative integer, and $e = 1.6 \times 10^{-19}\,C$.

Coulomb’s Law and the Electric Force

Charles Coulomb (1736–1806) established the mathematical relationship for the force between stationary charged particles using a torsion balance. 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 formula is:

$F = k \frac{|q_1 q_2|}{r^2}$

Where $k$ is a proportionality constant defined as:

$k = \frac{1}{4 \pi \epsilon_o \epsilon_r}$

In this expression, $\epsilon_o$ is the permittivity of free space, valued at $8.854 \times 10^{-12}\,C^2/Nm^2$ (or $F/m$), and $\epsilon_r$ is the relative permittivity of the medium. Relative permittivity is the ratio of the force between two charges in a vacuum to the force between the same charges in a specific medium. Absolute value bars are used because the force magnitude is always positive, even though charges can be negative. Coulomb's Law specifically applies to point charges. The directions of the forces are always along the line joining the two charges, and they obey Newton’s third law, being equal in magnitude and opposite in direction.

The Principle of Superposition

When multiple charges exert forces on a single charge simultaneously, the total force is the vector sum of the individual forces. This is known as the principle of superposition. For a collection of charges, the resultant force on charge $q_1$ by charges $q_2, q_3, q_4, \dots$ is given by:

$\mathbf{F}_1 = \mathbf{F}_{21} + \mathbf{F}_{31} + \mathbf{F}_{41} + \dots$

This principle allows for the calculation of complex electrostatic interactions by breaking them down into pairs of point charges. When calculating the resultant force on a charge in a multi-dimensional arrangement (such as charges at the corners of a triangle), the forces must be resolved into their $x$ and $y$ components before summing them to find the magnitude and direction of the net force.

Electric Field and Field Intensity

An electric field is the region of space surrounding a charged body where another charged particle experiences a force. The electric field $\mathbf{E}$ at a point is defined as the electric force $\mathbf{F}$ experienced by a test charge $q_o$ at that point, divided by the magnitude of the test charge:

$\mathbf{E} = \frac{\mathbf{F}}{q_o}$

The SI unit for electric field magnitude is newtons per coulomb ($N/C$). Electric field intensity is a vector quantity, and its direction is the same as the direction of the force exerted on a positive test charge. For a point charge $q_2$ at a distance $r$, the magnitude of the electric field is:

$E = k \frac{|q_2|}{r^2}$

Just like electric forces, electric fields follow the superposition principle. The total field at a point due to a group of charges is the vector sum of the fields produced by each individual charge:

$\mathbf{E} = \sum \mathbf{E}_n = k \sum \frac{|q_i|}{r_i^2}$

Electric Field Lines

Electric field lines, or lines of force, are imaginary lines used to visualize the features of an electric field. These lines are continuous and represent the field throughout space. For a positive point charge, the lines emanate radially outward, while for a negative charge, the lines are directed radially inward. Consequently, field lines siempre originate from positive charges and terminate on negative charges.

The properties of electric field lines include:

1. They start at positive charges and end at negative charges.
2. The tangent to a field line at any point indicates the direction of the electric field at that point.
3. Field lines are always normal (perpendicular) to the surface of the charged body at the point of origin or termination.
4. The density of the lines indicates field strength; lines are closer together where the field is strong and further apart where the field is weak. In a uniform electric field, the lines are parallel and equidistant.

Questions & Discussion

Question: There are two charged bodies, X and Y, which attract each other. X repels a third charged body Z. Do you think Z will attract or repel Y? Answer: Z will attract Y. Since X and Y attract, they have opposite charges. Since X and Z repel, they have the same charge. Therefore, Z and Y have opposite charges and will attract.

Question: What type of electric charge does a body that gains electron(s) possess? Answer: It possesses a negative charge because electrons carry negative charge.

Question: What are the properties of electric charge? Answer:

• Unlike charges attract and like charges repel.
• Net charge of an isolated system remains constant (Conservation).
• Charge is quantized ($q = ne$).
• There are two types: positive and negative.

Question: A conductor possesses a positive charge of $3.2 \times 10^{-19}\,C$. How many electrons does it have in excess or deficit? ($e = 1.60 \times 10^{-19}\,C$) Answer: It has a deficit of 2 electrons. Because the charge is positive, the body has lost electrons. Using $n = \frac{q}{e} = \frac{3.2 \times 10^{-19}}{1.6 \times 10^{-19}} = 2$.

Question: Two charges, one of $+5 \times 10^{-7}\,C$ and the other $-2 \times 10^{-7}\,C$, attract each other with a force of $-100\,N$. How far apart are they? Answer: Approximately $3\,mm$. (Calculation involves rearranging Coulomb's Law for $r$).

Question: Find the magnitude and direction of the resultant force $F_3$ in the right-triangle example where $q_1 = q_3 = 5.0\,\mu C$, $q_2 = -2.0\,\mu C$, and $a = 0.10\,m$. Answer: The resultant force is $8.0\,N$ at an angle of $98^{\circ}$ with the x-axis.

Question: Two equal positive charges $q_1 = q_2 = 2.0\,\mu C$ are at $(0, 0.30\,m)$ and $(0, -0.30\,m)$. What is the force on $Q = 4.0\,\mu C$ at $(0.40\,m, 0)$? Answer: The force from each charge is $0.29\,N$. The total force is in the $+x$ direction with a magnitude of $0.46\,N$.