Electromagnetism, Light, and Optics: A Comprehensive Overview
Huygens’ Principle
This principle describes a mechanism for constructing wavefronts. A wavefront is a surface passing through points where a wave oscillates with the same phase. Huygens’ principle states:
- Points on a wavefront act as sources of secondary waves.
- The envelope of these secondary waves forms a new primary wavefront.
To apply this principle, small circles (representing secondary waves) are drawn centered at points on the initial wavefront. The new wavefront is the envelope of these circles.
One consequence of Huygens’ principle is that all rays (lines perpendicular to wavefronts, indicating wave propagation direction) take the same time to travel between consecutive wavefronts. Although initially formulated for matter waves, Huygens’ principle applies to all types of waves, including electromagnetic waves (as later extended by Kirchhoff).
Electric Potential Energy and Electric Potential
The force between two charges is conservative, meaning it has an associated electric potential energy function (Ep). The difference in potential energy between two points equals the work done by the electric force between those points. Potential energy is typically set to zero at infinity and is measured in joules (J).
Under the influence of the electric force, a charge moves towards positions of minimum electric potential energy. The total electric potential energy of a set of charges is the sum of the potential energies of all possible pairs of charges.
The electric field is also conservative and has an associated scalar field called electric potential, measured in volts (V). The potential difference between two points is also known as voltage. The potential due to a set of charges is the scalar sum of the potentials due to each charge.
Electric Charge and Coulomb’s Law
Electric charge is a fundamental property of matter responsible for electromagnetic interaction. Its properties include:
- It can be positive or negative.
- The total charge of a group of particles is the sum of their individual charges (considering signs).
- The total electric charge of an isolated system is conserved.
- Charge is quantized, meaning it exists in discrete multiples of a fundamental charge (e = 1.6 x 10-19 C). Electrons have a charge of -e, and protons have a charge of +e. The SI unit of charge is the coulomb (C).
Coulomb’s law describes the force between stationary electric charges:
The force between two point charges (q1 and q2) is proportional to the product of the charges and inversely proportional to the square of the distance (r) between them. The force acts along the line connecting the charges and is repulsive for like charges and attractive for opposite charges.
where k = 1/(4πε0) = 9 x 109 Nm2/C2 is Coulomb’s constant, and ε0 is the permittivity of free space. The constant k has different values in media other than vacuum.
Electrostatic forces obey the superposition principle: the net force on a charge due to multiple other charges is the vector sum of the individual forces.
Lorentz Force
A charged particle (q) moving with velocity (v) in a magnetic field (B) experiences a force given by:
F = q(v x B)
This force is perpendicular to both the velocity and the magnetic field. Its magnitude is:
F = qvBsin(α)
where α is the angle between v and B. The force is maximum when v and B are perpendicular and zero when they are parallel or when the particle is at rest or has no charge.
Since the magnetic force is perpendicular to the particle’s path (and thus its velocity), it does no work (dW = F⋅dr = 0) and doesn’t change the particle’s kinetic energy, only its direction.
In a uniform magnetic field, the magnetic force acts as a centripetal force, causing the charged particle to move in a circle. The radius of this circle can be calculated from qvB = mv2/R.
If an electric field (E) is also present, the force on the charge is F = q(E + v x B), which is the general form of the Lorentz force.
Electromagnetic Induction
Electromagnetic induction is the process of generating electricity from magnetism. Key observations about induced currents in a circuit include:
- Relative motion between a magnet and the circuit.
- Relative motion between the circuit and another current-carrying circuit.
- A varying current in a second circuit, even if both circuits are stationary.
- Deformation of the circuit within a magnetic field.
The electromotive force (emf) is the work done per unit charge moved around the circuit, measured in volts (V). Magnetic flux (Φ = B⋅S, where S is the surface area) changes when the magnetic field, the circuit’s shape, or the orientation between the field and the circuit changes. These changes induce currents.
Faraday’s law states: emf = -dΦ/dt. The negative sign reflects Lenz’s law: the induced emf creates a current whose magnetic field opposes the change in magnetic flux. A key application of electromagnetic induction is generating electric current by converting mechanical work into electricity (e.g., hydroelectric power plants).
Nature of Light
Historically, two main theories competed to explain the nature of light:
- Particle theory (e.g., Newton): light consists of particles or corpuscles.
- Wave theory: light behaves as a wave.
Both theories explained reflection and refraction, but only the wave theory adequately explained interference, diffraction, and the fact that light travels faster in less dense media. Maxwell’s work on electromagnetism further supported the wave theory, establishing light as an electromagnetic wave in the 19th century.
However, in the early 20th century, Einstein invoked the particle nature of light to explain phenomena like the photoelectric effect. This led to the concept of wave-particle duality: light exhibits both wave-like and particle-like behavior. In some phenomena, light acts as an electromagnetic wave; in others, it acts as a stream of particles called photons.
Laws of Reflection and Refraction
When a wave encounters a boundary between two media with different refractive indices, part of the wave is reflected, and part is refracted (transmitted). The laws of reflection and refraction state:
- The incident, reflected, and refracted rays lie in the same plane (the plane of incidence), which is perpendicular to the surface.
- The angle of incidence (θi) equals the angle of reflection (θr).
- The angle of incidence and the angle of transmission/refraction (θt) are related by Snell’s law: n1sin(θi) = n2sin(θt), where n1 and n2 are the refractive indices of the first and second media.
Snell’s law implies that when light enters a medium with a higher refractive index, the rays bend towards the normal (and away from the normal when entering a medium with a lower index). Snell’s law can also be expressed in terms of the speed of light in the two media: n = c/v.
Lens Power and Focal Length
The focal length (f) is the distance from the lens to the image focus, where the image of an object at infinity is formed. The object focal length (f’) is the distance to the object focus, the point whose image forms at infinity. It is equal to the focal length with the opposite sign (f’ = -f). Converging lenses have positive focal lengths, and diverging lenses have negative focal lengths.
The power (P) of a lens is the reciprocal of the focal length, measured in diopters (D or m-1):
P = 1/f = -1/f’ = (nL – 1)(1/R1 – 1/R2)
where nL is the refractive index of the lens, and R1 and R2 are the radii of curvature of the lens’s surfaces. This formula assumes the lens is in air. The power is positive for converging lenses and negative for diverging lenses, depending on the values and signs of the radii of curvature.