SME
University Physics III: Waves, Optics & Modern · Problem type
Single-slit diffraction
Use this for “Red light of 650 nm passes a slit 0.05 mm wide onto a wall 2.5 m away. How wide is the central bright band?”
Diagram
First
dark
fringe:
sin
θ₁
=
λ
a
=
500
nm
0.1
mm
=
0.005,
θ₁
=
0.2865°
Central band: w = 2L tan θ₁ = 2 × 1.5 m × tan 0.2865° = 15 mm, twice each side band.
Across is not to scale: the screen is much farther than drawn.
This is an example. Type your own values to start a new problem.
a (mm)
sin(
θ₁
°)
=
λ (nm)
w (mm)
=
2
×
L (m)
tan(
θ₁
°)
sin θ₁
Sine of the first dark angle
worked out (not typed)
Formulas
sin
θ₁
=
λ
a
×
10
6
0.005
=
500
0.1 ×
10
6
θ₁
=
sin⁻¹(sin
θ₁
)
0.2865 = sin⁻¹(0.005)
w
=
2
×
L
×
tan(
θ₁
)
×
1,000
15.0002 = 2 × 1.5 × tan(0.2865°) × 1,000
Assumptions
•
One wavelength, and the screen is far compared with a, so the rays from the slit to one point are parallel.
•
Dark fringes where a sin θ = mλ, with m = ±1, ±2, …; the central bright band runs between the first two, at ±θ₁.
•
The central band is w = 2L tan θ₁ wide, about twice each side band; for small angles w ≈ 2λL ÷ a.
•
λ is in nm and a in mm, so a is a × 10⁶ nm; w comes out in mm (× 1000).
Step-by-step
WE KNOW
λ = 500 nm, a = 0.1 mm, L = 1.5 m
FIND
sine of the first dark angle (sin θ₁), angle to the first dark fringe (θ₁), central band width (w)
Step 1 · Find sine of the first dark angle (sin θ₁)
Use sin θ₁ = λ ÷ (a × 10⁶)
From
a
sin
θ₁
=
λ
(m
=
1):
the
wavelength
over
the
slit
width,
both
in
nm.
sin
θ₁
=
500
0.1 ×
10
6
sin
θ₁
=
0.005
Step 2 · Find angle to the first dark fringe (θ₁)
Use θ₁ = sin⁻¹(sin θ₁)
The angle whose sine that is.
θ₁
=
sin⁻¹(0.005)
θ₁
=
0.28648°
Step 3 · Find central band width (w)
Use w = 2 × L × tan(θ₁) × 1,000
The
first
dark
fringes
sit
L
tan
θ₁
either
side
of
the
centre:
twice
that,
in
m,
then
×
1000
for
mm.
w
=
2
×
1.5
×
tan(0.28648°)
×
1,000
w
=
2
×
1.5
×
0.005
×
1,000
w
=
3
×
0.005
×
1,000
w
=
0.015
×
1,000
w
=
15.0002
mm
Check
Put the numbers back in. Both sides should match.
0.005
=
500
0.1 ×
10
6
✓
0.28648 = sin⁻¹(0.005) ✓
15.0002 = 2 × 1.5 × tan(0.28648°) × 1,000 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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