Electromagnetic Induction & Maxwell Equations

Week 1 - Electromagnetic Induction

• According to Faraday's Law, the magnitude of induced electromotive force (EMF) relies on the rate of change of magnetic flux.
• Induced EMF increases if the magnetic flux through a coil changes rapidly.
• Increasing the speed of the conductor's motion also increases the magnitude of induced EMF.
• Eddy currents, generators, and transformers are application for electromagnetic induction.

Week 1 - Maxwell's Evaluation

• An electrostatic field arises when electrons are stationary (hence, "static"), producing a corresponding electric potential.
• Electrostatic field is caused by the changing flux in the conductor.
• The cylindrical symmetry allows the inference that the electric field has a constant magnitude throughout the circle and is tangent to the circle at any given point.
• An electric guitar is an application of a magnetically induced electric field.
• The guitar contains stationary coils that induce electric currents through the vibration of the guitar's magnetic strings.

Week 1 - Faraday's Law

• A solenoid with a radius of 0.01 m contains 20 turns of the coil.
• The solenoid's uniform magnetic field changed from 30.0 × 10-3 T in one direction to 30.0 × 10-3 T in the opposite direction.
• The time interval is 45 × 10-3 s.
• The induced EMF is -8.37 × 10-3 V.

Week 1 - Lenz's Law

• The induced current will generate its own magnetic field in the case of a changing magnetic flux in a motionless or stationary circuit.
• Induced current opposes the direction of the change in magnetic flux through the loop.
• When the induced magnetic field opposes the flux change upward, the induced current direction is counterclockwise.
• When the induced magnetic field opposes the flux change downward, the induced current direction is clockwise.

Week 2 - LC Circuits in AC and DC

• AC (alternating current) and DC (direct current) are two types of electric current.
• Alternating current electrons move back and forth, while direct current electrons move in one direction.
• "Alternating Current" (AC) is used in a typical household because alternating current travels farther than DC current.
• Capacitors and inductors complement each other because they form a circuit with inductive and capacitive components, which is called is a tuned circuit.
• The capacitor becomes charged with electrical energy when connected to the solenoid.
• Electrical impedance is the measure of the opposition or resistance in an AC circuit.

Week 2 -Electromagnetic Waves and Light

• An electromagnetic (EM) wave is composed of time-varying magnetic and electric fields at right angles, traveling at 2.99 x 10^8 m/s in free space; EM waves oscillate in a sinusoidal wave shape.
• Frequency and wavelength are two properties of EM waves with an inverse relationship.
• Visible light comprises a short fraction of electromagnetic radiation that we see with our eyes.

Week 2 - Law of Refraction

• The two conditions that account for total internal reflection include:
  • Light must travel from a denser to a less dense medium (water to air and glass to air).
  • The angle of incidence must be greater than the critical angle.
• A material's refractive index is determined by its optical density of a material's molecules to keep the absorbed energy of an EM wave in the form of oscillating electrons before releasing it back as a new disturbance.
• Snell's Law relates the ratio of sines of angles to the inverse ratio of refractive indicies: nₐsinθₐ = nᵦsinθᵦ

Week 2 - Dispersion

• Rainbows are seen because light rays from the sun refract as they pass through the air to water droplets in the atmosphere.
• The formation of rainbows by dispersion is explained by the same physics responsible for the multi-color splitting of light as it passes through a prism, known as prismatic effect.
• The refractive index and light wavelength, according to Snell's Law, has an inverse relationship.

Week 3 - Polarization

• For an ideal analyzer transmitting initially unpolarized light with initial intensity 4I₀, the polarized light intensity after passing through the analyzer is 2I₀.
• When initially unpolarized light of intensity 4I₀ passes through a polarizer, the transmitted intensity is reduced by a factor of 1/2.
• The intensity after the polarizer is Iafter polarizer = (1/2) × 4I₀ = 2I₀.
• The analyzer (a second polarizer) is placed after the first polarizer.
• The intensity of the transmitted light is given by Malus’ Law Iₐ = Iₚcos²θ
• The final intensity of the transmitted light after the analyzer depends on the angle θ between the polarizer and the analyzer I = 2I₀cos²θ
• If the analyzer is aligned with the first polarizer (θ=0°), the maximum intensity is 2I₀.
• If the analyzer is perpendicular to the first polarizer (θ = 90°), no light is transmitted.
• A laser beam travels in the air through ethyl alcohol at an angle of incidence of 25.15°, then the angle of refraction is 18.22°, where nₐ = 1 and n = 1.36.

Week 4 - Image Formation

• Geometric optics is the specific discipline of studying light as a ray.
• Spherical mirrors are curved mirrors that are either concave or convex.
• Concave mirrors can form both real and virtual images.
• Convex mirrors always form a virtual, erect, and diminished image.
• Spherical aberration is the defect that occurs when the outer parts of a mirror do not bring light rays into a similar focus as that of the central part.

Week 4 - MIrrors and Lenses

• The principal axis is an imaginary line passing through the center of the sphere from where the mirror is cut from to the vertex of the spherical mirror
• Focal length is the distance between the vertex and the focal point.
• A lens is a piece of transparent material, glass or plastic, shaped to bend light rays in a specific way as they pass through it.

Week 4 - Images Formed by a Plane Mirror

• The precise location of a virtual image (P’) as formed by a plane mirror:
  • Is behind the mirror.
  • Is laterally inverted.
  • Is the same size as the object.
  • Is virtual.
  • Is erect.

Week 4 - Images Formed by Lenses (Graphical Method)

• When the object is at a finite distance beyond the focal point (s>2f), the image formed is real inverted, diminished, and located between the focus and 2F on the other side.

Week 4 - Application of Geometric Optics

• The human eye is one of the most powerful optical instruments.
• To ensure clear vision, the eye lens changes its refractive power to focus a real image on the retina for objects at varying distances.
• Any object closer to the near point (25 cm) will have a blurry image.
• A Snellen chart identifys if a person has normal or ideal vision.
• Nearsightedness or myopia happens when a nearsighted person cannot accommodate images that are over the near point of 25 cm.
• Diverging lenses are used to correct nearsightedness.

Week 5 - Huygen's Principle and the Double-Slit experiment

• When a laser is beamed into a very narrow slit onto a screen, light diffracts into the geometrical shadow, causing the light waves to overlap; This effect that affects the pattern resulting from the diffraction on the screen is called the superposition principle.
• Laser Lights display real effects of interference.
• Thomas Young, the scientist credited for the double-slit experiment, is a fundamental physics experiment that demonstrates the wave-particle duality of light and matter and has since become one of the most important experiments in quantum mechanics.
• Interference occurs when waves overlap to combine additively (constructively) or subtractively (destructively) through a process called superposition.
• Constructive interference occurs when two waves are in phase and their amplitudes add up
• Destructive interference occurs when two waves are out of phase and their amplitudes subtract.

Week 5 - Single-Slit Experiment

• A slow increase, rather than a sudden one, in the intensity of light occurs as it changes from dark to light.
• When light is monochromatic, diffraction fringes at the boundaries of the shadow are created.
• Longer wavelengths create more diffracted light.
• Mirage is produced when a light ray hits a straight edge.
• the diffraction fringes become wider in a diffraction pattern.

Supplementary notes:

• Astigmatism occurs when the cornea of the eye is not spherical.
• Oftentimes ophthalmologists prescribe cylindrical lenses to correct astigmatism.
• A cornea is a refractive medium.