Revision: Wave Theory of Light Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

Alignment of dipole moments (permanent or induced) in the direction of an applied electric field is called polarisation.

The variation in intensity of sound with time at a particular position, due to the principle of superposition of two sound waves of slightly different frequencies, is called beats.

The periodic variation of intensity of sound between maximum and minimum due to superimposition of two sound waves of same amplitude and slightly different frequencies is called the phenomenon of beats.

The maximum intensity point produced during the formation of beats is called waxing.

The minimum intensity point produced during the formation of beats is called waning.

Defined as dipole moment per unit volume:

\[P=\frac{\text{dipole moment}}{\mathrm{volume}}=np\]

The number of beats formed per second is expressed as ∣v₁ − v₂∣, i.e., either (v₁ − v₂) or (v₂ − v₁), where v₁​ and v₂ are frequencies of the two sound waves.

N = n₁ ​− n₂​

The beat period is the reciprocal of beat frequency:

T = \[\frac{1}{n_1-n_2}\] or T = \[\frac{1}{|v_1-v_2|}\]

Statement:

When unpolarised light is incident at polarising angle i_B on an interface separating air from a medium of refractive index μ, then the reflected light is plane polarised (perpendicular to the plane of incidence), provided:

Additional condition at polarising angle:

i.e., the reflected plane polarised light is at right angles to the refracted light.

OR

Statement:

• When the angle of incidence equals the polarising angle (θ_B), the reflected and refracted rays are perpendicular to each other.
• "The refractive index of a medium is equal to the tangent of the polarising angle θ_B."

Consider two sound waves, having the same amplitude and slightly different frequencies n₁ and n₂. Let us assume that they arrive in phase at some point x of the medium. The displacement due to each wave at any instant of time at that point is given as

`y_1 = A sin {2pi (n_1t - x/lambda_1)}`

`y_2 = A sin {2pi (n_2t - x/lambda_2)}`

Let us assume for simplicity that the listener is at x = 0.

∴ y₁ = A sin (2πn₁t)     ...(i)

and y₂ = A sin (2πn₂t)     ...(ii)

According to the principle of superposition of waves,

y = y₁ + y₂

∴ y = A sin (2πn₁t) + A sin (2πn₂t)

By using formula,

sin C + sin D = 2 sin `((C + D)/2) cos ((C − D)/2)`

y = `A[2sin((2pin_1t + 2pi n_2t)/2 )] cos [((2pin_1t - 2pin_2t)/2)]`

y = `2A sin [2pi ((n_1 + n_2)/2)t] cos [2pi ((n_1 - n_2)/2)t]`

∴ y = `R sin [2pi ((n_1 + n_2)/2)t]`

y = R sin (2πnt)     ...(iii)

Where,

R = `2A cos[(2pi(n_1 - n_2))/(2)t]` and n = `(n_1 + n_2)/2`

Equation (iii) is the equation of a progressive wave having frequency `((n_1 + n_2)/2)` and resultant amplitude R.

For waxing,

A = ± 2a

`therefore 2A cos [2pi((n_1 - n_2)/2)t] = +- 2A`

`therefore cos [2pi ((n_1 - n_2)/2)]t = +-( 2A)/(2A)`

`therefore cos [2pi ((n_1 - n_2)/2)]t = +- 1`

This is possible if

`2pi ((n_1 - n_2)/2)t = 0, pi, 2pi, 3pi, ....`

i.e. t = 0, `1/(n_1 - n_2), 2/(n_1 - n_2), 3/(n_1 - n_2), ...`

∴ Period of beat T = `[1/(n_1 - n_2) - 0]`

T = `1/(n_1 - n_2)`

∴ Frequency of beats n = `1/T`

n = n₁ − n₂

Thus, the frequency of beats is equal to the difference between the frequencies of the two sound notes giving rise to beats.

• Every point on a wavefront acts as a secondary source (point source) that emits new spherical wavelets in all directions with the same speed as the original wave.
• The new (forward) wavefront at any later time is the common tangential envelope (tangent surface) to all these secondary wavelets.
• The wavefront in a medium is always perpendicular to the direction of wave propagation.
• Secondary wavelets travel only in the forward direction — backward wavelets are ignored (zero amplitude in backward direction).

Memory: Every point → new source → envelope = new wavefront.

• Beats are formed when two waves of same amplitude but slightly different frequencies superimpose.
• Waxing and waning are alternatively produced.
• The greater the difference in frequency between the two waves, the higher the beat frequency.