5.2 Coulomb's law

Coulomb's_Law_0### is the foundation of electrostatics, describing the force between charged particles. It quantifies how charges interact, showing that the force is proportional to charge magnitudes and inversely proportional to distance squared.

Understanding Coulomb's law is crucial for grasping electric fields, potential, and energy. It connects to broader electromagnetic theory and has practical applications in technology, from microscopy to photocopying, highlighting its importance in modern physics and engineering.

Coulomb's law formula

• Describes the electrostatic force between two point charges
• Fundamental equation in electrostatics that quantifies the interaction between electrically charged particles
• Mathematically expressed as $F = k \frac{q_1 q_2}{r^2}$, where $F$ is the force, $k$ is Coulomb's constant, $q_1$ and $q_2$ are the magnitudes of the charges, and $r$ is the distance between them

Electric force between charges

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• Directly proportional to the product of the magnitudes of the charges ($q_1$ and $q_2$)
• Inversely proportional to the square of the distance ($r$) between the charges
• Attractive force for opposite charges (positive and negative) and repulsive force for like charges (both positive or both negative)

Magnitude and direction

• Magnitude of the force determined by the absolute values of the charges and the distance between them
• Direction of the force along the line connecting the two charges
  • Attractive force points toward the opposite charge
  • Repulsive force points away from the like charge

Units of electric charge

• Measured in coulombs (C) in the International System of Units (SI)
• One coulomb is the charge transferred by a current of one ampere in one second ($1\,\text{C} = 1\,\text{A} \cdot 1\,\text{s}$)
• Charges can be positive or negative, with the elementary charge ($e$) being the smallest unit of charge (approximately $1.602 \times 10^{-19}\,\text{C}$)

Coulomb's constant

• Proportionality constant in Coulomb's law, denoted by $k$
• Represents the strength of the electrostatic force between charges in a given medium

Value in SI units

• In vacuum or air, $k = 8.988 \times 10^9\,\text{N} \cdot \text{m}^2/\text{C}^2$
• Often expressed in terms of the permittivity of free space ($\varepsilon_0$) as $k = \frac{1}{4\pi\varepsilon_0}$, where $\varepsilon_0 \approx 8.854 \times 10^{-12}\,\text{F}/\text{m}$

Dependence on medium

• Value of $k$ depends on the medium in which the charges are placed
• In a dielectric medium, $k$ is reduced by a factor of the relative permittivity ($\varepsilon_r$) of the medium: $k = \frac{1}{4\pi\varepsilon_0\varepsilon_r}$
• Relative permittivity is a dimensionless quantity that characterizes the electric polarizability of the medium (for vacuum, $\varepsilon_r = 1$)

Electric field of point charges

• Region around a point charge in which it exerts an electric force on other charges
• Electric field strength ($\vec{E}$) is the force per unit charge: $\vec{E} = \frac{\vec{F}}{q}$
• Measured in units of newtons per coulomb (N/C) or volts per meter (V/m)

Field lines and direction

• Electric field represented by field lines, which are imaginary lines that show the direction of the electric force on a positive test charge
• Field lines originate from positive charges and terminate on negative charges
• Density of field lines indicates the strength of the electric field (denser lines correspond to stronger fields)

Superposition principle

• Electric field generated by multiple point charges is the vector sum of the individual fields created by each charge
• Allows for the calculation of the resultant electric field at any point in space by adding the contributions from all charges
• Mathematically, for $n$ point charges: $\vec{E}_{\text{total}} = \sum_{i=1}^{n} \vec{E}_i$, where $\vec{E}_i$ is the electric field due to the $i$-th charge

Force between multiple charges

• Coulomb's law can be extended to calculate the force on a charge in the presence of multiple other charges
• Principle of superposition applies to electric forces as well

Pairwise force calculation

• Force on a charge due to multiple other charges is the vector sum of the individual pairwise forces
• Each pairwise force is calculated using Coulomb's law, considering the magnitude and direction of the force between the two charges

Net force on a charge

• Resultant force on a charge is the vector sum of all the pairwise forces acting on it
• Mathematically, for $n$ charges: $\vec{F}_{\text{net}} = \sum_{i=1}^{n} \vec{F}_i$, where $\vec{F}_i$ is the force due to the $i$-th charge
• Net force determines the acceleration and motion of the charge according to Newton's second law ($\vec{F}_{\text{net}} = m\vec{a}$)

Limitations of Coulomb's law

• Coulomb's law is a fundamental principle in electrostatics but has certain limitations in its applicability

Applicable to static charges

• Coulomb's law is valid for charges that are at rest or moving with constant velocity
• Accurately describes the electrostatic force between stationary point charges

Invalid for moving charges

• Coulomb's law does not account for the magnetic fields generated by moving charges
• For charges moving with accelerated motion, the electromagnetic force must be described by the more general Lorentz force law ($\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$)
• Relativistic effects must be considered for charges moving at high velocities

Coulomb's law vs Newton's law

• Coulomb's law for electrostatic force and Newton's law of universal gravitation share similarities in their mathematical formulation

Similarities in formulas

• Both laws describe inverse-square force laws, where the force is inversely proportional to the square of the distance between the interacting objects
• Gravitational force: $F_g = G \frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant and $m_1$ and $m_2$ are the masses of the objects
• Electrostatic force: $F_e = k \frac{q_1 q_2}{r^2}$, where $k$ is Coulomb's constant and $q_1$ and $q_2$ are the magnitudes of the charges

Differences in forces

• Gravitational force is always attractive, while electrostatic force can be attractive (opposite charges) or repulsive (like charges)
• Strength of the gravitational force is much weaker than the electrostatic force (gravitational constant $G \approx 6.674 \times 10^{-11}\,\text{N} \cdot \text{m}^2/\text{kg}^2$ is much smaller than Coulomb's constant $k \approx 8.988 \times 10^9\,\text{N} \cdot \text{m}^2/\text{C}^2$)
• Gravitational force acts on mass, while electrostatic force acts on electric charge

Applications of Coulomb's law

• Coulomb's law has numerous practical applications in various fields, including physics, engineering, and technology

Electric force microscopy

• Technique used to image surface topography and measure local electric properties at the nanoscale
• Relies on the electrostatic force between a conductive tip and the sample surface
• Coulomb's law is used to interpret the force measurements and determine the local electric charge distribution

Electrostatic precipitation

• Method for removing particulate matter (dust, smoke) from air or gas streams
• Particles are charged by ions and then attracted to oppositely charged collection plates due to the electrostatic force
• Coulomb's law governs the attraction between the charged particles and the collection plates

Xerography and photocopying

• Process used in photocopiers and laser printers to create images on paper
• Photoconductor drum is charged, and light exposure selectively dissipates the charge in the image areas
• Toner particles are attracted to the remaining charged areas on the drum due to the electrostatic force described by Coulomb's law
• Toner is then transferred and fused onto the paper to create the final image