Electricity_and_Magnetism_24-25

Electricity and Magnetism

Electrostatics

• Introduction: Electrostatics is the systematic study of the forces, fields, and potential of electric charges at rest. Conversely, magneto-statics studies the forces and fields of steady current.

Coulomb’s Law

• Fundamental Principle: Like charges repel and unlike charges attract each other, resulting in an electrical force.

• Calculation of Force: This electrical force can be quantified using Coulomb’s law.

Statement of Coulomb’s Law

• Mathematical Formulation: Coulomb's law expresses that the electrical force (F) between two point charges is directly proportional to the product of their charges (q1, q2) and inversely proportional to the square of the distance (r) between them:

  [ F = k \frac{q_1 q_2}{r^2} ]

• Value of k: The constant k can be defined in terms of the permittivity of free space (ε0=8.85418782×10^-12 m^-3 kg^-1 s4 A2).

Vector Nature of Force

• Vector Quantity: Force is a vector quantity, necessitating direction alongside magnitude.

• Example Forces:

  • Force on charge Q2:

  1. F2 = 2103 K a12

  2. Force on charge Q1: F1=9102 K A21

Superposition Principle

• Multiple Charges: For more than two point charges, the net force acting on any single charge is the vector sum of the forces acting on it due to all other charges. This is determined using the principle of superposition:

  [ F_{3} = F_{13} + F_{23} ]

Problem-Solving Steps on Coulomb’s Law

1. Determine the position vectors of the charges.

2. Calculate the unit vector connecting the charges to ascertain the direction of the force applied.

3. Apply Coulomb’s law in vector form to express the forces involved.

4. If applicable, repeat for additional charges affecting the target charge until all forces are calculated.

5. Sum all forces vectorially to derive the resultant force acting on the charge.

Practical Problems

Problem 1:

• Two like charges at a distance of d = 5 cm exerting a force of F = 9 × 10^-3 N. Calculate:(a) Magnitude of each charge.(b) Direction of the electrostatic force between them.

Problem 2:

• Given a point charge of 4 µC and -1 µC separated by 3 cm. Calculate the magnitude of the Coulombic force and determine whether it is attractive or repulsive.

Problem 3:

• For three charged particles (A = -5 μC, B = +10 μC, C = -12 μC) arranged linearly, compute the net electrostatic force on particle B resulting from particles A and C’s influence.

Continuous Charge Distribution

Types

• Linear Charge Distribution: Charges uniformly distributed along a length denoted by λ, defined as ( \lambda = \frac{dq}{dl} ), measured in Coulombs per meter (C/m).

• Surface Charge Distribution: Charges are uniformly distributed over a given area, denoted by σ, defined as ( \sigma = \frac{dq}{da} ), measured in Coulombs per square meter (C/m²).

• Volume Charge Distribution: Here, charges are uniformly distributed within a volume denoted by ρ, defined as ( \rho = \frac{dq}{dV} ), measured in Coulombs per cubic meter (C/m³).

Electric Field

• Definition: Electric fields surround charged particles and exert a force on other charges. Denoted as E, its strength represents the force on a unit positive charge in an electric field.

• Calculating Electric Field Intensity: Defined as the force per unit charge exerted on a test charge.

Units of Electric Field

• Units: Electric field strength is expressed in N/C (Newtons/Coulomb) and V/m (Volts per meter).

Electric Field Lines

• Characteristics:

  • Field lines never intersect.

  • They are perpendicular to charged surfaces, indicating field strength through density.

  • The proximity of lines indicates field strength—closer lines denote stronger fields, while further lines indicate weaker fields.

Applications of Electric Field Calculations

• The concept allows calculations like fields due to linear charge distributions and uniformly charged objects.

Electrostatic Potential

• Defined as the work done in moving a unit charge from infinity to a point in the field. The relation ( V = \frac{W}{q} ) captures this potential energy conversion.

Electric Flux

• Electric flux measures the number of electric field lines crossing a surface. Expressed as:

  • ( \Phi = E \cdot S ) (where S is the area normal to the field).

  • If the surface is oriented at an angle, it modifies to: ( \Phi = (E \cos(θ)) S ).

Magnetic Considerations

Magnetostatics

• Magnetic fields arise due to steady currents and are characterized by their magnetic lines of force, indicating the presence and influence of magnetic effects.

Biot-Savart Law

• Determines the magnetic field generated at a point due to a current-carrying conductor, establishing a relationship between the elements of current, their vector direction, and how they contribute to the magnetic field.

Ampere’s Law

• Establishes that the line integral of the magnetic field along a closed path correlates with the current encircled by the path:[ oxed{\oint B , dl = \mu_0 I} ]

Gauss’s Law for Magnetism

• States that the total magnetic flux through any closed surface is zero, underpinning the absence of magnetic monopoles and reinforcing the conservation of magnetic field lines.

Maxwell’s Equations

• Represent the synthesis of electromagnetism, detailing how electric and magnetic fields propagate. They highlight electric displacement, magnetic monopoles, the electromotive force, and conduction currents in dynamic fields.

Significance of Maxwell’s Equations

1. Total electric displacement corresponds to total charge within volume.

2. Magnetic monopoles do not exist.

3. The electromotive force and magnetic displacement interrelate.

Applications of Electromagnetism

• Utilized in technologies like MRI scanning, GPS, wireless communication, and electric motor functions, underscoring the significance and influence of electromagnetic communications in modern techniques.

Waveguides and Modes

• Structures guiding electromagnetic waves are pivotal in telecommunications and related technologies, with various propagation modes contributing to their versatility in applications.

Conclusion

Electromagnetism, alongside electric and magnetic fields, plays a crucial role in the fundamental understanding and application of physics in technology and daily life.