1.4 Coulomb’s Law of Electrostatics and superposition Principle

1.4.1 Coulomb’s law

Coulomb’s law describes the electrostatic force between two-point charges.  The superposition principle allows calculating net forces or fields from multiple charges by vector addition.

Coulombs law state that; the magnitude of the electrostatic force between two-point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically expressed F =   

Where:

F = electrostatic force (N)

Q1 and Q2 = magnitudes of the charges (C)

R = distance between the charges (M)

K = Coulomb’s constant (9.0 x  N.m²/C²)

Note: The force is Attractive for unlike charges and Repulsive for like charges.

Work examples:

1.      An electron is at a distance of 5.0x m away from the center of the proton. Calculate the electrostatic force on it.?

2.      A point charge of +3.0× C is 12.0cm distant from a second point charge of −1.50× C. Calculate the magnitude of the force on each charge.

3.      What must be the distance between point charge q1=26.0µC and point charge q2=          47.0µC for the electrostatic force between them to have a magnitude of 5.70N

1.4.2 Superposition Principle

Superposition Principle State that for multiple charges, the net force on any charge is the vector sum of forces from each charge individually. It is given as Fnet = . Similarly, they can superpose  while potentials add scalarly:

The linearity systems from Maxwell’s equations and simplifies complex systems as see in prior Gauss’s law discussion for symmetric fields.

Example calculations

1.      A circuit consist of two DC voltage sources of 12V and 6V connected in series with the resistor of 6ohm. Using the superposition principle determine the current following through the resistor

2.      A resistor of 4ohm is connected to a pair current source supplying current of 5A and 3A respectively in the same direction. Calculate the superpose voltage across the resistor.

3.      Two voltage sources of 10V and 4V are connected in opposite directions across a 2ohm resistor. Calculate the net current in the circuit.

4.      A circuit contains a 20V voltage source and an 8V voltage connected with 4ohm resistor. Determine the current contributed by each source and hence calculate the total current in the resistor.

5.      In a certain electrical circuit, a 15V sources and a 5V source are connected in different directions across a 5ohm resistor.

6.      Determine a 3ohm resistor connected in a circuit containing two voltage sources of 9V and 3V.

7.      Two current sources of 2A and 6A are connected to a 5ohm resistor.

1.5 Electric Field

A charged object experiences a force in an electric field. This is what an electric field is. We say that there is an electric field anywhere an electric charge experiences a force. An electric field is a field of force.

An electric field is a region of sp ace around a charged particle or object in which another electric charge experiences an electric force.

Electric Field Strength at a point is defined as the electric force per unit positive test charge at a point in the field. E =  

Where:

E = electric field strength (N C⁻¹ or V m⁻¹)

F = electric force N

Q = test charge (C)

The electric field is a vector quantity, meaning it has both magnitude and direction.

The direction of the electric field at any point is the direction of the force acting on a positive test charge placed at that point.

1.6  Potential

Potential difference (p.d.), also called voltage, is the work done or energy transferred per unit charge in moving a charge between two points in an electric circuit. V = W / Q

Where: -

V = potential difference (V)

W = work done or energy transferred(J)

Q = charge (C)

1.7 Gauss’s Law and applications

Gauss’s law is a fundamental principle in electromagnetism that relates electric flux through a closed surface to the enclosed electric charge. It simplifies calculations for symmetric charges distributions.

 1.7.1 Gauss’s Law

This law state that the total electric flux through any closed surface (Gaussian Surface) equals the net charge (Qenc.) enclosed divided by the permittivity of Free space.

5.7.2 Applications of Gauss’s law

Gauss’s law excels for symmetric charge distributions, deriving fields without direct integration; This case derived Coulomb’s law and explain shielding e.g. Faraday cage

1.      Spherical symmetry: For instance, in a charge sphere, inside a uniformly charged hollow sphere E is zero; outside E, matches a point charge at the center.

2.      Cylindrical symmetry: For instance, infinite line charge

3.      Planar Symmetry: e.g. Infinite sheet, charge density

4.      Conductors: here field inside is zero; surface charge creates field only inside