Interference

• Constructive interference: path difference δ = m

• Destructive interference: path difference δ = (m + ½ )  (m = 0, 1, 2, 3, …)

Young’s Double Slit Experiment

The initial experiment setup consisted of a candle, as light source, and two sets of pinholes, which produced coherent light waves.

• Light emitted from large source (e.g. light bulb) is not coherent since emission is generally a random process. Each particle emits a “wave burst” at a different time.

• However, light from a point source is coherent. One can produce coherent light by letting light waves pass through a very small opening. The opening acts as a single secondary point source (Huygens principle).

• Laser (Light Amplification by Stimulated Emission of Radiation) sources are coherent as all particles in a material are stimulated to produce wave bursts at the same time

• Light from the same wave front is incident on both slits. The slits become coherent point sources.

• I nterference occurs on a distant screen, as alternating bright and dark bands, called fringes.

• The central fringe ( = 0) appears as a result of constructive interference for all wavelengths, therefore it will be white if white light is used.

For bright fringe

For dark fringe

• For D >> d >> λ the angle θ is small.

• Prove that the distance between the central maximum and the m^th fringe is:

Diffraction occurs when a wave is incident onto an obstacle, aperture or sharp edge. The wave “bends” around the obstacle or “spreads out” after passing through an aperture.

Huygens’s principle

Every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the waves.

S ingle slit diffraction

A single slit placed between a distant light source and a screen produces a diffraction pattern;

Diffraction grating

• A diffraction grating consists of a very large number of identical, equidistant, closely spaced parallel lines or slits on glass, metal or plastic;

• The pattern on the screen is produced through both interference and diffraction. Each slit or line produces diffraction, and the diffracted beams interfere with one another to form the final diffraction pattern;

• 

Summary

Laws of Reflection

• T he incident ray makes an angle θ₁ with the normal;

• The reflected ray makes an angle of θ₁’ with the normal

• Law 1:

• Law 2 : The incident ray, reflected ray and the normal are in the same plane

Refraction of Light

O ccurs when a ray of light from one medium passes through a boundary leading into another medium;

Part of the energy is reflected and part is refracted (enters the second medium);

The refracted ray that enters the second medium is bent at the boundary

(This bending of the ray is called refraction)

The reason why light changes direction (refracts) is due to fact that light travels at a different speed in different mediums;

The speed of light in any material is less than its speed in vacuum;

This property of a medium is described by the absolute refractive index of the medium.

Properties of light inside a medium

• The frequency remains unchanged

• The speed is reduced by a factor of n

• The wavelength is reduced by a factor n

Snell’s Law of Refraction

Wave moving from medium 1 to medium 2:

Prism

A monochromatic ray of light incident on the prism will emerge at an angle δ from its original direction of travel

• δ is called the angle of deviation;

• Φ is the apex angle.

Total internal reflection

• θ_r > θ_i when rays pass from a medium of higher refractive index into one of lower refractive index; however θ_r cannot exceed 90°;

• So, there is a limiting θ_i above which no refracted ray can form. This limiting angle is the critical angle θ_c;

• For θ_i > θ_c rays are reflected back into the first medium. This phenomenon is known as total internal reflection.

Special Relativity

Reference Frame (RF): the coordinate system in which measurements are made;

Inertial or Galilean RF: nonaccelerating RF in which Newton’s laws are valid;

Newtonian principle of Relativity

If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system.

Galilean Transformation

• Parallel axes;

• S' has a constant relative velocity in the x-direction with respect to S;

• Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers.

Einstein’s Postulates for the Special Theory of Relativity

1. Relativity Principle: “The laws of Physics are the same in all inertial frames of reference”

2. Constancy of Speed of Light: “The speed of light is constant (in a vacuum) in all inertial frames of reference, regardless of the observer’s or the light source’s motion”

Events in SR

• In any reference frame, an event is a physical situation, which is described by both position and time (x, y, z, t);

• In general, events that are simultaneous in one reference frame will not be simultaneous in other reference frames;

• Simultaneity depends on the observer’s reference frame or the observer’s state of motion.

Proper time, is characteristic to events occurring at the same coordinates system (RF).

Coordinate time, is characteristic to events occurring in different coordinates system (RF).

Two types of distance

L_P is called proper distance and it is measured by the observer at rest relative to the departure and destination points.

L_C is called coordinate distance and it is measured by the observer who moves at speed v relative to the departure and destination points.

This equation describes length contraction, as well; however, length contraction, unlike

distance contraction, is not a direct consequence of time dilation in the reference frame

at rest. In other words, a different proof is required for length contraction, which is (far)

beyond the scope of this course.

• An observer moving uniformly relative to another object, but parallel to that object’s length, measures a shorter length than an observer who is at rest relative to that object.

OR

• If an observer at rest measures the length of a moving object whose length is parallel to its direction of motion, then this length will be shorter than that measured by an observer (at rest) in the moving object’s reference frame.

Two types of mass: rest mass and relativistic mass

The relativistic mass of an object increases with increasing relative speed v, according to:

m_rest is measured in the reference frame in which both the observer and the object are at rest (i.e. v = 0 m/s);

m_rel is measured/observed when the object moves at v ≠ 0 relative to the observer or the observer moves at v ≠ 0 relative to the object.

Relativistic Momentum

Consequently, other physical quantities that depend on momentum will also depend on γ and hence v.

Energy Considerations: Relativistic KE

By considering the work done by a force F on a mass, initially at rest (m_rest), it can be shown that the relativistic Kinetic Energy takes this form:

Energy Considerations: Rest Energy

m_rest c² is the only component of KE_rel that has units of energy and it does not depend on the relative velocity (γ has no units).

Hence the rest mass is equivalent to an amount of energy E_rest that is inherent and due to the rest mass:

Energy Considerations: Total Energy

Total Energy, Rest Energy and Momentum

The following expression can be obtained by squaring the total energy and the relativistic momentum:

Mass and energy are equivalent: E = mc²