Slit Width & Single Slit Diffraction

Homework Statement

Consider ultrasound with a frequency of 40 kHz.

(a) What is the wavelength and period of a 40 kHz sound wave?
(b) The slit width required to give the first order diffraction minimum at thirty degrees.
(c) Calculate the angles at which the first three minimums should be observed for single slit diffraction with the following slit widths:
1. three and a half times the wavelength:
2. five times the wavelength:
3. twice the wavelength:

a sin θ = λ
sin θ = λ/a

The Attempt at a Solution

(a)

40kHz = 40,000Hz
v = speed of sound = 343m/s
f = 40,000Hz
f = 1/T
T = 1/f = 1/40,000 = [2.5*10^(-5)]
λ = v/f = 343/40000 = [8.575*10^(-3)nm]

I'm not sure if this is correct.

(b)

slit width = a
slit separation = d
angle = θ

a sin θ = λ
sin θ = λ/a
sin 30 = (8.575*10^(-3)) / a
0.5 = (8.575*10^(-3)) / a
a = λ/sin 30 = [0.01715um]

I am sure that this is wrong.

(c)

I am not sure how to calculate this with the limited variables that I have to work with.

ehild
Homework Helper

Homework Statement

Consider ultrasound with a frequency of 40 kHz.

(a) What is the wavelength and period of a 40 kHz sound wave?
(b) The slit width required to give the first order diffraction minimum at thirty degrees.
(c) Calculate the angles at which the first three minimums should be observed for single slit diffraction with the following slit widths:
1. three and a half times the wavelength:
2. five times the wavelength:
3. twice the wavelength:

a sin θ = λ
sin θ = λ/a

The Attempt at a Solution

(a)

40kHz = 40,000Hz
v = speed of sound = 343m/s
f = 40,000Hz
f = 1/T
T = 1/f = 1/40,000 = [2.5*10^(-5)]
λ = v/f = 343/40000 = [8.575*10^(-3)nm]

I'm not sure if this is correct.

(b)

slit width = a
slit separation = d
angle = θ

a sin θ = λ
sin θ = λ/a
sin 30 = (8.575*10^(-3)) / a
0.5 = (8.575*10^(-3)) / a
a = λ/sin 30 = [0.01715um]

I am sure that this is wrong.

(c)

I am not sure how to calculate this with the limited variables that I have to work with.
The period and wavelength are correct, but you need to include the units.
You gave the formula for the first diffraction minimum, what is it for other orders?
There is a mistake in your value of a.

The period and wavelength are correct, but you need to include the units.
You gave the formula for the first diffraction minimum, what is it for other orders?
There is a mistake in your value of a.

I am good with a) and b), but still struggling with c).

My lecturer says I need the diffraction equation from b) and to work out the inverse sine function. I understand what he is saying, but the question seems strangely worded.

(c) Calculate the angles at which the first three minimums should be observed for single slit diffraction with the following slit widths:
1. three and a half times the wavelength:
2. five times the wavelength:
3. twice the wavelength:

I am being asked to calculate NEW angles, that explains why I use the inverse sine function.

but then the question states '...with the following slit widths:', then proceeds to give multipliers of the wavelength. Which one is it? What do I keep constant?

ehild
Homework Helper

I am good with a) and b), but still struggling with c).

My lecturer says I need the diffraction equation from b) and to work out the inverse sine function. I understand what he is saying, but the question seems strangely worded.

(c) Calculate the angles at which the first three minimums should be observed for single slit diffraction with the following slit widths:
1. three and a half times the wavelength:
2. five times the wavelength:
3. twice the wavelength:

I am being asked to calculate NEW angles, that explains why I use the inverse sine function.

but then the question states '...with the following slit widths:', then proceeds to give multipliers of the wavelength. Which one is it? What do I keep constant?

You have to determine those angles for each slit widths.
The equation you wrote for the first minimum is
a sin(θ)=λ
If a= 3.5λ,
3.5λsin(θ)=λ.
What is sin(θ) ? What is θ?
What is θ if a= 5λ? if a=2λ?
What are the equation for the second and third minimums?