Double Slit

Question 1

What are the conditions for interference to occur?

Answer 1

• sources must be coherent
• ordinary sources have randomly fluctuating phase, two points illuminated by the source will only be coherent if they are separated by less than the coherence length

Question 2

Coherent

Definition

Answer 2

-sources are coherent if they maintain a constant phase difference

Question 3

Young’s Double Slit Experiment

Description

Answer 3

• illuminate two very narrow slits by a small distant source
• each slit behaves as a source
• get patches of constructive and destructive interference

Question 4

What does phase difference depend on?

Answer 4

path difference

Question 5

Young’s Double Slit Experiment

Path Difference

Answer 5

-to reach a point P, light from the two slits has to travel different distances
-as the distance to the screen L ->∞ and the paths become approximately parallel
Δ = d sin(θ)
d = distance between slits

Question 6

Phase Difference Equation

Answer 6

ϕ = 2πΔ/λ

Question 7

Young’s Double Slit Experiment

Phase Difference

Answer 7

ϕ = 2π/λ * d sin(θ)

Question 8

Young’s Double Slit Experiment

Constructive Interference

Answer 8

-occurs when the signals are in phase
i.e. when ϕ = 2πm
where m = 0, ±1, ±2, ±3, …
-bright fringes are therefore found when the path difference is Δ = mλ

Question 9

Young’s Double Slit Experiment

Destructive Interference

Answer 9

-occurs when the signals are out of phase
i.e. when ϕ = 2π(m + 1/2)
where m = 0, ±1, ±2, ±3, …
-dark fringes therefore occur when Δ = (m + 1/2) λ

Question 10

Young’s Double Slit Experiment

Zeroth Order Maximum

Answer 10

-the central maximum
m = 0
θ = 0

Question 11

Young’s Double Slit Experiment

Small Angle Approximation

Answer 11

sinθ ≈ θ ≈ y/L

• can be used if the distance across the screen is smaller than the distance between the slits and the screen
  i. e. if y

Question 12

Young’s Double Slit Experiment

ym Equation

Answer 12

ym = m * λL/d

• where ym is the distance along the screen from the zeroth order maximum to the mth bright fringe
• this is only true for small angles

Question 13

Young’s Double Slit Experiment

Distance Between Bright Fringes

Answer 13

ym+1 - ym = λL/d

-hence for small angles the fringes are equally spaced on the screen

Question 14

Young’s Double Slit Experiment

Superposition of Electric Fields

Answer 14

-the electric field at a point P is a superposition of the fields due to the two sources s1 and s2
Etot = E1 + E2
-E1 and E2 are equal in frequency and amplitude but differ in phase by ϕ
-so E1 = E0 sin(ωt)
and E2 = E0 sin(ωt + ϕ)
Etot = 2E0(sin(ωt)+sin(ωt + ϕ))
giving
Etot = 2E0cos(ϕ/2)sin(ωt+ϕ/2)
-so Etot has same frequency as each source but an amplitude of 2E0cos(ϕ/2)

Question 15

Young’s Double Slit Experiment

Intensity Distribution

Answer 15

light intensity ∝ (field amplitude)²
I ∝ (2E0cos(ϕ/2))²
-so I = 4I0cos²(ϕ/2)
-where I0 = E0², is the intensity due to a single source
-the maxima occur when the phase difference is an integer multiplied by 2π

Question 16

Interference and Energy

Answer 16

-interference does not create or destroy energy, it just redistributes it