Comprehensive Notes on Electrostatics and Coulomb's Law

Fundamental Principles of Electrostatics

• Definition of Electrostatics:

  • Electrostatics is the study of electric charges that are at rest or static.

  • The electric force between static charged particles is known as the electrostatic force.

  • The electrostatic force is an attractive or repulsive force between particles caused due to their electric charges. It is also referred to as Coulomb's force.

• Unit and Quantization of Charge:

  • In SI units, electric charge is measured in coulombs (symbol $C$).

  • The charge $q$ is quantized and exists as an integral multiple of the elementary charge $e$:         $q = n e$

• Circuit Application:

  • Capacitance and its dependence on dielectrics represent another major quantity playing an important role in electrical circuits.

Coulomb's Law

• Historical Context:

  • The quantitative measurement of the force between two electric charges was first made by Charles-Augustin de Coulomb (1736–1805), an eminent French physicist.

  • Coulomb carried out a series of experiments using an apparatus known as a torsion balance to measure the force between electric charges.

  • He expressed his experimental data as Coulomb's law.

• Statement of Coulomb's Law:

  • Coulomb's law states that "the magnitude of the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them."

• Mathematical Formulation:

  • For two electric charges $q_1$ and $q_2$ separated by a distance $r$:

    • Force is directly proportional to product of charges:             $F \propto q_1 q_2$

    • Force is inversely proportional to square of distance:             $F \propto \frac{1}{r^2}$

    • Combining proportionalities:             $F \propto \frac{q_1 q_2}{r^2}$

    • Scalar force magnitude equation:             $F = k \frac{q_1 q_2}{r^2}$

  • $k$ is a constant of proportionality whose value depends upon the system of units used and the medium between the charges.

  • Point Charge Condition: The electric charges $q_1$ and $q_2$ are assumed to be point or localized charges, provided that the size of the bodies carrying the charges is very small compared to the distance $r$ between them.

• Vector Form and Newton's Third Law:

  • To specify direction, unit vectors along the line joining the two charges are used.

  • If $\hat{r}_{12}$ is a unit vector pointing from charge $q_1$ towards charge $q_2$, the force $\vec{F}_{21}$ exerted by charge $q_1$ on $q_2$ in vector form is:         $\vec{F}_{21} = k \frac{q_1 q_2}{r^2} \hat{r}_{12}$

  • If $\hat{r}_{21}$ is a unit vector pointing from charge $q_2$ towards charge $q_1$, the force $\vec{F}_{12}$ exerted by charge $q_2$ on $q_1$ in vector form is:         $\vec{F}_{12} = k \frac{q_1 q_2}{r^2} \hat{r}_{21}$

  • Since $\hat{r}_{12} = -\hat{r}_{21}$, combining the equations yields:         $\vec{F}_{21} = -\vec{F}_{12}$

  • Conventionally, $\vec{F}_{21}$ denotes force exerted by $q_1$ on $q_2$, and $\vec{F}_{12}$ denotes force exerted by $q_2$ on $q_1$.

  • The force $\vec{F}_{12}$ exerted by $q_2$ on $q_1$ is equal in magnitude and opposite in direction to $\vec{F}_{21}$ exerted by $q_1$ on $q_2$, proving that Coulomb's law fits into Newton's third law.

Permittivity and Electrostatics in Material Media

• Free Space Permittivity:

  • When charges are separated by air or vacuum, constant $k$ is expressed in terms of the permittivity of free space $\varepsilon_0$:         $k = \frac{1}{4 \pi \varepsilon_0}$

  • Experimentally measured value of constant $k$:         $k = 9.0 \times 10^9\,N\,m^2\,C^{-2}$

  • Value of permittivity of free space $\varepsilon_0$:         $\varepsilon_0 = 8.85 \times 10^{-12}\,C^2\,N^{-1}\,m^{-2}$

• Influence of Insulators (Dielectrics):

  • When an insulator is placed between electric charges, experimental observation shows that it reduces the force.

  • Permittivity is defined as the property of a medium which affects the magnitude of force between two point charges.

  • Coulomb's force in a medium with permittivity $\varepsilon$:         $F_{med} = \frac{1}{4 \pi \varepsilon} \frac{q_1 q_2}{r^2}$

• Relative Permittivity (Dielectric Constant):

  • The permittivity of a material medium compared with the permittivity of vacuum is called relative permittivity or dielectric constant $\varepsilon_r$:         $\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}$

  • $\varepsilon_r$ is a dimensionless constant, and its value is always greater than unity ($\varepsilon_r > 1$) for various dielectrics.

  • Force in a medium with relative permittivity $\varepsilon_r$ in vector form:         $\vec{F}_{med} = \frac{1}{4 \pi \varepsilon_0 \varepsilon_r} \frac{q_1 q_2}{r^2} \hat{r}$

  • Relation between force in medium and force in vacuum ($F_{vac}$):         $F_{med} = \frac{F_{vac}}{\varepsilon_r}$

• Relative Permittivity ($\varepsilon_r$) Values Table:

  • Vacuum: $1$

  • Air: $1.0006$

  • Benzene: $2.284$

  • Germanium: $16$

  • Water: $78.5$

  • Glass: $4.8 - 10$

  • Mica: $3 - 7.5$

  • Paraffine paper: $2$

  • Rubber: $2.94$

  • Ammonia (liquid): $22 - 25$

Principle of Superposition and Example Problem

• Superposition of Multiple Charges:

  • When multiple charges act on a single charge, the net resultant force is the vector sum of individual forces.

  • If five charges are placed such that charges $q_2, q_3, q_4, q_5$ exert forces on charge $q_1$, the resultant force $\vec{F}_1$ is:         $\vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \vec{F}_{14} + \vec{F}_{15}$

• Worked Example 11.1:

  • Problem Statement: Three charges $q_1 = -5\,\mu C$, $q_2 = +10\,\mu C$, and $q_3 = -12\,\mu C$ are placed in a line. Calculate the net electrostatic force on charge $q_2$ due to the other two charges.

  • Given Data:

    • Charge $q_1 = -5\,\mu C = -5 \times 10^{-6}\,C$

    • Charge $q_2 = +10\,\mu C = +10 \times 10^{-6}\,C$

    • Charge $q_3 = -12\,\mu C = -12 \times 10^{-6}\,C$

    • Constant $k = 9 \times 10^9\,N\,m^2\,C^{-2}$

    • Distance between charge $q_1$ and $q_2$ ($r_{12}$) = $6\,cm = 0.06\,m = 6 \times 10^{-2}\,m$

    • Distance between charge $q_2$ and $q_3$ ($r_{23}$) = $4\,cm = 0.04\,m = 4 \times 10^{-2}\,m$

  • Required:

    • Magnitude of net electrostatic force on charge $q_2 = F_{net} = ?$

    • Direction of electrostatic force = ?

Electric Field and Its Intensity

• Concept of Electric Field:

  • The concept of an electric field was first proposed by Michael Faraday in the 19th century.

  • Faraday stated that the electric field around a charge is like a sphere within which other charges are influenced by it.

  • An electric field is defined as any region around a charge in which an electric test charge would experience an electric force.

• Properties and Determination of Field:

  • An electric field is characterized by strength and direction at every point in space.

  • The strength and direction of an electric field are determined by placing a unit positive test charge in that field.

  • The direction in which this unit positive test charge moves or tends to move is the direction of the electric field.

  • The test charge $q_0$ is so small that it does not distort the original field due to the primary source.

• Electric Field Intensity ($\vec{E}$):

  • A single vector quantity containing information about the field strength and its direction at a given point is denoted by $\vec{E}$ and is known as electric field intensity.

  • If a unit positive test charge $q_0$ experiences a force $\vec{F}$ due to the electric field of charge $q$, the intensity of an electric field at any point is the force per unit positive test charge placed at that point:         $\vec{E} = \frac{\vec{F}}{q_0}$

  • In force form, the relation is written as:         $\vec{F} = q_0 \vec{E}$