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Learn/ 11th Physics/ Unit 8 /Numericals

Unit 8: Physical Optics and Gravitational Waves — Numericals

11th Class Physics · Unit 8: Physical Optics and Gravitational Waves

8.1.When an unpolarized light of intensity I₀ is incident on a polarizing sheet, find the intensity of light which does not get transmitted.
Given
Intensity of unpolarized light I0
Formula
Intensity transmitted by polarizer I_{transmitted = I02
Polarizer blocks half of intensity (polarizer blocks half of intensity of light)
Intensity not transmitted I_{not transmitted = I0 - I_{transmitted
Working
Substituting I_{not transmitted = I0 - I02 = I02
I02
8.2.A polarized light beam passes through a polarizer at an angle of 45°. Find the intensity of the transmitted light if the initial intensity is 100 W m⁻².
Given
Initial intensity I0 = 100 W/m2
Angle θ = 45^circ
Formula
Malus's Law I = I0 cos2 θ
Working
Substituting values I = 100 × cos2(45^circ)
Computing cos(45°) I = 100 × left(frac{1}{sqrt{2}right)2 = 100 × 0.5
Result
Transmitted intensity I = 50 W/m2
8.3.A light wave passes through a polarizer with its electric field aligned at 30° to the horizontal. If the amplitude of the wave is 10 units, what is the amplitude of the wave passing through the polarizer?
Given
Initial amplitude A0 = 10 units
Angle θ = 30^circ
Formula
Amplitude after polarizer (Malus's Law) A = A0 cos(θ)
Working
Substituting values A = 10 × cos(30^circ)
Computing cos(30°) A = 10 × 0.866
Result
Amplitude after polarizer A = 8.66 units
8.4.What angle is required between the direction of polaroid light and the axis of a Polaroid filter to reduce its intensity by 85%?
Given
Initial intensity I = I0
Percentage decrease in intensity 85%
New intensity after 85% reduction I = I0 - 85 I0100
Simplifying I = 0.15 I0
Formula
Malus's Law I = I0 cos2 θ ⇒ II0 = cos2 θ
Working
Substituting 0.15 = cos2 θ
Taking square root cos θ = sqrt{0.15} = 0.387
Finding angle θ = cos-1(0.387) = 67.5^circ
Result
Required angle θ = 67.5^circ
8.5.An unpolarized light having intensity of 15 W m⁻² is incident on a pair of polarizers. The first polaroid filter has its transmission axis at 50° from the vertical. The second Polaroid filter has its transmission axis at 20° from the vertical. Calculate the intensity of light transmitted to both filters.
Given
Intensity of unpolarized light I0 = 15 W m-2
Angle of 1st polarizer from vertical θ1 = 50^circ
Angle of 2nd polarizer from vertical θ2 = 20^circ
Formula
Intensity after 1st polarizer (unpolarized light) I1 = I02
Working
Substituting I1 = 152 = 7.5 W m-2
Angle between 1st and 2nd polarizers θ = 50^circ - 20^circ = 30^circ
Formula
Intensity after 2nd polarizer (Malus's Law) I2 = I1 cos2 θ
Working
Substituting I2 = 7.5 × cos2(30^circ)
Computing cos(30°) I2 = 7.5 × left(frac{sqrt{3}{2}right)2 = 7.5 × 0.75
Result
Intensity after both filters I2 = 5.625 W m-2
8.6.Two polarizing sheets have their polarizing directions parallel so that intensity of emitted light is maximum. Through what angle must either sheet be rotated if the intensity is to be dropped by half?
Given
Initial intensity I0
Intensity of transmitted light initially I_{transmitted = I02
Formula
After rotation by angle θ I = I0 cos2 θ
Working
Setting up equation for half intensity I02 = I0 cos2 θ
Simplifying 12 = cos2 θ
Taking square root cos θ = sqrt{12 = frac{1}{sqrt{2}
Finding angle θ = cos-1left(frac{1}{sqrt{2}right) = 45^circ
θ = 45^circ
8.7.We wish to use a glass plate of refractive index of 1.5 in air as a polarizer. Find the polarizing angle and angle of refraction.
Given
Refractive index of glass n_{glass = 1.5
Refractive index of air n_{air = 1
Formula
Brewster's angle formula tan θp = frac{n_{glass{n_{air
Working
Substituting θp = tan-1(1.5)
Computing angle θp = tan-1(1.5) = 56.3^circ
Formula
Angle of refraction θr = 90^circ - θp
Working
Substituting θr = 90^circ - 56.0^circ = 33.7^circ
θp = 56.3^circ; θr = 33.7^circ
8.8.At what angle of incidence, will light reflect from water be completely polarized?
Given
Refractive index of water n_{water = 1.33
Formula
Brewster's angle formula tan θp = n_{water
Working
Substituting θ = tan-1(1.33)
Computing angle θ ≈ 53^circ
θ ≈ 53^circ
8.9.A beam of unpolarized light is incident on a stack of four polarizing sheets that are lined up so that the characteristic direction of each is rotated by 30° clockwise with respect to the preceding sheet. What fraction in percentage of the incident intensity be transmitted?
8.10.A polarizer and an analyzer have their axes aligned at 60°. What is the fraction of the initial intensity that emerges?
Given
Initial intensity I0
Angle between polarizer and analyzer θ = 60^circ
Formula
Using Malus's Law I = I0 cos2 θ
Working
Substituting I = I0 × cos2(60^circ) = I0 × 0.25
Fraction of intensity II0 = 0.25
Fraction = 0.25
8.11.If the gravitational waves have a wavelength of 3000 km, then find their frequency assuming it moves with the speed of light?
Given
Wavelength lambda = 3000 km = 3 × 106 m
Speed of light c = 3 × 108 m/s
Formula
Wave equation v = f lambda
Frequency formula f = clambda
Working
Substituting values f = 3 × 1083 × 106 = 102
Result
Frequency f = 100 Hz