Electrostatics: (Coulomb’s Law, Electric Field, Potential)
Explore Coulomb’s Law, Electric Fields, and Potential Energy
Introduction to Electrostatics
Electrostatics studies the phenomena arising from stationary or slow-moving electric charges. This fundamental branch of physics explains how charged particles interact with each other and forms the basis for understanding electricity, magnetism, and many modern technologies.
Key Concepts in Electrostatics:
- Electric charge is a fundamental property of matter
- Like charges repel; opposite charges attract
- Charge is conserved in isolated systems
- Charge is quantized (exists in discrete amounts)
- Charge interactions follow the inverse-square law
The study of electrostatics begins with understanding electric charge. Electric charge comes in two varieties: positive and negative. The most common carriers of electric charge are protons (positive) and electrons (negative). The SI unit of electric charge is the coulomb (C), named after Charles-Augustin de Coulomb.
One coulomb is approximately equal to the charge of 6.24 × 1018 electrons. The elementary charge (e) is 1.602 × 10-19 C, which is the magnitude of charge carried by a single electron or proton.
Coulomb’s Law
Coulomb’s Law describes the electrostatic force between two point charges. Formulated by French physicist Charles-Augustin de Coulomb in 1785, this fundamental law states that 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.
F = ke · |q1q2| / r2
where F is the electrostatic force, ke is Coulomb’s constant (8.99 × 109 N·m2/C2), q1 and q2 are the magnitudes of the charges, and r is the distance between them.
Properties of Coulomb’s Law:
- Vector Nature: The force is a vector quantity, acting along the line joining the two charges.
- Attractive or Repulsive: The force is repulsive if the charges have the same sign and attractive if they have opposite signs.
- Inverse-Square Relationship: The force decreases with the square of the distance, similar to Newton’s law of gravitation.
- Superposition Principle: The net force on a charge due to multiple charges is the vector sum of individual forces.
Coulomb’s Constant in Different Media
In a medium other than vacuum, Coulomb’s constant is modified by the relative permittivity (εr) of the medium:
ke = 1/(4πε0εr)
where ε0 is the permittivity of free space (8.85 × 10-12 F/m) and εr is the relative permittivity of the medium.
Interactive Coulomb’s Law Demonstration
Click to explore how force changes with charge and distance
Electric Field
An electric field is a region around a charged particle or object within which a force would be exerted on other charged particles or objects. The electric field concept was introduced by Michael Faraday to explain how forces act at a distance.
E = F/q = ke · Q / r2
where E is the electric field, F is the force experienced by a test charge q, Q is the source charge, and r is the distance from the source charge.
The SI unit of electric field is newton per coulomb (N/C) or volt per meter (V/m), where 1 N/C = 1 V/m.
Electric Field Lines
Electric field lines provide a visual representation of the electric field in space. These imaginary lines show the direction that a positive test charge would move if placed in the field.
Properties of Electric Field Lines:
- Start from positive charges and end on negative charges
- Never cross each other
- Closer lines indicate stronger fields
- Always perpendicular to equipotential surfaces
Electric Field Due to Multiple Charges
The electric field due to multiple charges follows the superposition principle. The total electric field at any point is the vector sum of the electric fields due to individual charges:
Etotal = E1 + E2 + E3 + … + En
Electric Field of Common Charge Distributions
| Charge Distribution | Electric Field Expression |
|---|---|
| Point charge | E = keQ/r2 |
| Infinite line charge (linear charge density λ) | E = 2keλ/r |
| Infinite plane (surface charge density σ) | E = σ/(2ε0) |
| Conducting sphere (radius R, charge Q) | E = keQ/r2 (r > R), E = 0 (r < R) |
Electric Potential
Electric potential is the amount of work needed to move a unit positive charge from a reference point (usually infinity) to a specific point against an electric field. It’s a scalar quantity that provides an alternative way to describe electric fields.
V = ke · Q / r
where V is the electric potential, Q is the source charge, and r is the distance from the source charge.
The SI unit of electric potential is the volt (V), where 1 volt equals 1 joule per coulomb (J/C).
Relationship Between Electric Field and Potential
The electric field is the negative gradient of the electric potential:
E = -∇V
In one dimension: E = -dV/dx
This relationship means that the electric field points in the direction of decreasing potential, similar to how water flows downhill.
Equipotential Surfaces
Equipotential surfaces are surfaces where the electric potential is constant. Key properties include:
- No work is done when moving a charge along an equipotential surface
- Electric field lines are always perpendicular to equipotential surfaces
- For a point charge, equipotential surfaces are concentric spheres
- The surface of a conductor in electrostatic equilibrium is an equipotential surface
Electric Potential Energy
When a charge q is placed in an electric potential V, it possesses electric potential energy:
U = q · V
The potential energy of a system of point charges is the work required to assemble the configuration.
For two point charges q1 and q2 separated by distance r:
U = ke · q1q2 / r
Interactive Potential Visualization
Click to explore equipotential surfaces and electric field lines
Real-world Applications
Electrostatics principles are applied in numerous technologies and natural phenomena. Understanding these applications helps connect theoretical concepts to practical scenarios.
Technology Applications
- Electrostatic Precipitators: Remove particles from gas streams in industrial settings and power plants
- Photocopiers and Laser Printers: Use electrostatic attraction to transfer toner to paper
- Electrostatic Painting: Efficiently coats objects by attracting charged paint particles
- Capacitors: Store electric charge and energy in electronic circuits
- Van de Graaff Generators: Produce high voltages for research and demonstrations
Natural Phenomena
- Lightning: Discharge of static electricity accumulated in clouds
- Static Cling: Attraction between clothes due to charge separation in dryers
- Cell Membranes: Maintain potential differences crucial for cellular function
- Nerve Impulses: Propagate through changes in electric potential
- Gecko Adhesion: Geckos stick to surfaces partly due to electrostatic forces
Electrostatics Safety Considerations
Understanding electrostatics is crucial for safety in many industries:
- Grounding systems to prevent charge buildup
- Anti-static measures in electronics manufacturing
- Lightning protection systems for buildings
- Electrostatic discharge (ESD) protection for sensitive components
- Safety procedures for fuel handling to prevent sparks
Solved Examples
Example 1: Coulomb’s Law
Problem: Two point charges, q1 = 3.0 × 10-6 C and q2 = -2.0 × 10-6 C, are separated by a distance of 0.15 m in vacuum. Calculate the electrostatic force between them.
Solution:
Using Coulomb’s law: F = ke · |q1q2| / r2
F = (8.99 × 109 N·m2/C2) × |(3.0 × 10-6 C) × (-2.0 × 10-6 C)| / (0.15 m)2
F = (8.99 × 109) × (6.0 × 10-12) / (0.0225)
F = 2.4 N
Since the charges have opposite signs, the force is attractive, directed along the line joining the charges.
Example 2: Electric Field
Problem: A point charge of 5.0 × 10-9 C is located at the origin. Calculate the electric field at point P(0, 3.0 m).
Solution:
The distance from the charge to point P is r = 3.0 m
Using the electric field formula: E = ke · Q / r2
E = (8.99 × 109 N·m2/C2) × (5.0 × 10-9 C) / (3.0 m)2
E = (8.99 × 109) × (5.0 × 10-9) / 9.0
E = 5.0 N/C
The electric field at point P is 5.0 N/C, directed along the positive y-axis (away from the positive charge).
Example 3: Electric Potential
Problem: Calculate the electric potential at a distance of 0.5 m from a point charge of 2.0 × 10-8 C. What is the potential difference between points at 0.5 m and 1.0 m from the charge?
Solution:
Using the electric potential formula: V = ke · Q / r
At r = 0.5 m:
V0.5 = (8.99 × 109 N·m2/C2) × (2.0 × 10-8 C) / (0.5 m)
V0.5 = 359.6 V
At r = 1.0 m:
V1.0 = (8.99 × 109 N·m2/C2) × (2.0 × 10-8 C) / (1.0 m)
V1.0 = 179.8 V
Potential difference: ΔV = V0.5 – V1.0 = 359.6 V – 179.8 V = 179.8 V
The potential difference between the two points is 179.8 V.
Frequently Asked Questions
What is the difference between electric field and electric potential?
+Electric field is a vector quantity that represents the force per unit charge at a point in space. It indicates both magnitude and direction of the force that would act on a positive test charge. Electric potential, on the other hand, is a scalar quantity representing the potential energy per unit charge. While electric field shows how a charge would move, electric potential shows the energy associated with that position. Mathematically, the electric field is the negative gradient of the electric potential.
Why does Coulomb’s law have an inverse-square relationship?
+Coulomb’s law follows an inverse-square relationship because electric field lines spread out in three dimensions from a point charge. As the distance increases, these field lines distribute over an increasingly larger spherical surface area. Since the surface area of a sphere is proportional to the square of its radius (4πr²), the strength of the electric field (and thus the force) decreases with the square of the distance. This geometric spreading is a fundamental property of fields that emanate from a point source in three-dimensional space.
Can electric field lines cross each other?
+No, electric field lines cannot cross each other. If they did, this would mean that at the point of intersection, the electric field would have two different directions simultaneously, which is physically impossible. The electric field at any point in space has a unique direction, which is the direction of the net force that would act on a positive test charge placed at that point. This is why electric field lines never intersect.
What happens to the electric field inside a conductor?
+In electrostatic equilibrium, the electric field inside a conductor is zero. When an electric field is applied to a conductor, the free electrons redistribute themselves until they create an electric field that exactly cancels the external field inside the conductor. This happens because if there were a net electric field inside, it would cause the free charges to move, contradicting the assumption of electrostatic equilibrium. This property is used in Faraday cages to shield sensitive equipment from external electric fields.
How are MKS units used in electrostatics equations?
+In the MKS (meter-kilogram-second) system, which is part of the SI system, electrostatics equations use the following units:
- Force (F): newton (N = kg·m/s²)
- Charge (q): coulomb (C)
- Distance (r): meter (m)
- Electric field (E): newton per coulomb (N/C) or volt per meter (V/m)
- Electric potential (V): volt (V = J/C)
- Coulomb’s constant (ke): newton-square meter per coulomb-squared (N·m²/C²)
Consistent use of these units ensures that calculations yield correct results with proper physical dimensions.