HC Verma Solutions Class 12 Chapter 22 X-Rays offers detailed answers to all the exercises given in the chapter. Students can, therefore, use the solution to not only know the correct answers but they will get a clear understanding of the right method to solve these questions. The solutions will further help them develop better problem-solving skills and ultimately boost their preparation level. The solutions are centred around physics topics such as frequency, momentum and the energy of X-rays, etc. Students will also get to learn topics of X-Rays in problems such as:
- Finding the electric field when wavelength and diameter are given.
- Finding energy utilised by X-Rays when the wavelength is given.
- Questions based on energy needed to knock out an electron from their shells and energy needed to ionize an atom, etc.
- We will be seeing questions in which we have to find the maximum potential difference without emitting an electron, the power emitted in X-ray, heat produced, stopping potential and momentum of the atom.
The HC Verma solutions which have been thoughtfully prepared will also help students to be familiar with the concepts and practice the questions in a seamless manner.
Key Topics Related To X-rays
Some of the key topics that students should focus on are;
- Production of X-rays
- Continuous and Characteristic X-rays
- Soft and Hard X-Rays
- Moseley’s Law
- Bragg’s Law
- Properties and Uses of X-rays
Class 12 Important Questions In Chapter 22
1. State whether X-rays can be used or not for photoelectric effect? Is it possible for X-rays to be polarized?
2. Electrons usually strike the target and stop inside a Coolidge tube. Explain whether the target gets negatively charged as time passes in the process?
3. It is often said that X-ray is injurious to health. However, exposure to visible light is not harmful. Explain why this happens when both of them are electromagnetic waves?
4. What is the nature of the X-ray beam coming from an X-ray tube?
(a) It is monochromatic (b) the wavelengths are smaller than a certain maximum wavelength (c) the wavelengths are greater (d) All the wavelengths lie between the minimum and maximum wavelength
5. What happens to the emitted radiation when the potential difference applied to an X-ray tube is increased?
(a) the minimum wavelength decreases (b) the minimum wavelength increases (c) the intensity increases (d) there is no change in the level of intensity
HC Verma Solutions Vol 2 X-Rays Chapter 22
Question 1: Find the energy, the frequency and the momentum of an X-ray photon of wavelength 0.10 nm.
Wavelength of X-rays = 0.10 nm.
We know, E = hc/λ
Velocity of light = c = 3 x 108 m/s
Planks constant = h = 6.63 x 10-34 J-s
Wavelength of X-rays = λ = 0.10 x 10-9 m (given)
E = [6.63 x 10-34 x 3 x 108 ]/[ 0.10 x 10-9] = 1.98 x 10-15 Joules/photon
Also, the frequency can be calculated using below formula:
γ = c/ λ = [3 x 108]/[ 0.10 x 10-9] = 3 x 1018 Hertz
p = h/ λ = [6.63 x 10-34]/[ 0.10 x 10-9] = 6.626 x 10-24 kg ms-1.
Question 2: Iron emits Kα X-ray of energy 6.4 keV and calcium emits Kα X-ray of energy 3.69 keV. Calculate the times taken by an iron Kα photon and a calcium Kα photon to cross through a distance of 3 km.
Let “t” be the time taken each of those rays to travel a distance of 3 Km.
We know, time = distance/speed
=> t = [3×103]/[3×108] = 10-5 s = 10 μs
Both the Kα photon and X-ray photon will take the same i.e. 10μs
Question 3: Find the cutoff wavelength for the continuous X-rays coming from an X-ray tube operating at 30 kV.
Cutoff wavelength = λ = hc/eV = [1242 eV-nm]/[ex30x103] = 414 x 10-4 nm
Question 4: What potential difference should be applied across an X-ray tube to get X-ray of wavelength not less than 0.10 nm? What is the maximum energy of a photon of this X-ray in joule?
We know, λ = hc/eV
Or V = hc/eλ = [6.63 x 10-34 x 3 x108]/[1.6×10-19x10-10] = 12.4 KV
Maximum energy of photon of wavelength λ = 0.10 nm is
E = hc/λ = [6.63 x 10-34 x 3 x108]/[0.10×10-9] = 1.98 x 10-15 Joules/photon
Question 5: The X-ray coming from a Coolidge tube has a cutoff wavelength of 80 pm. Find the kinetic energy of the electrons hitting the target.
We know, E = hc/λ = [1242 eV-nm]/[80×10-3] = 15.525 x 103 eV = 15.5 KeV(approx..)
Question 6: If the operating potential in an X-ray tube is increased by 1%, by what percentage does the cut-off wavelength decrease?
If the operating voltage is increased by 1%, then the new operating voltage, say v’ will be,
V’ = V + V/100 = 1.01 V
Cutoff wavelength (λ’) in increasing the operation voltage is
λ’ = (h/1.01) V = λ/1.01
Therefore, difference in wavelength is λ’ – λ/1.01 = (0.01)/1.01λ
Percentage change in the wavelength:
(0.01)/1.01λ x λ x 100 = 1/1.01 = 0.9901 = 1 % (approx)
Question 7: The distance between the cathode (filament) and the target in an X-ray tube is 1.5 m. If the cutoff wavelength is 30 pm, find the electric field between the cathode and the target.
The distance between the cathode (filament) and the target in an X-ray tube is 1.5 m. If the cutoff wavelength is 30 pm.
We know, E = hc/ λ
= [6.63 x 10-34 x 3 x108]/[30×10-12] = 41.4 x 103 eV
Electric field = V/d = [41.4 x 103]/1.5 = 27.6 kVm-1.
Question 8: The short-wavelength limit shifts by 26 pm when the operating voltage in an X-ray tube is increased to 1.5 times the original value. What was the original value of the operating voltage?
When the operating voltage of the X-ray tube is increased to 1.5 times
we have, λ’ = λ – 26
where λ = initial wavelength and λ’ = new wavelength
Also, we have E = hc/λ
or eV = hc/λ
Where V = operating initial potential
λ = hc/eV
Consider V’ be the new operating voltage, then
λV = λ’V’ = (λ – 26) x 1.5 V
=> 0.5λ = 26 x 1.5
=> λ = 78 pm
Question 9: The electron beam in a color TV is accelerated through 32 kV and then strikes the screen. What is the wavelength of the most energetic X-ray photon?
E = hc/ λ
Or λ = hc/E
Given: Potential across the X-ray Tube, V = 32kV
=> λ = hc/E = [1242×10-9]/[32×103] = 38.8 pm
Question 10: When 40 kV is applied across an X-ray tube, X-ray is obtained with a maximum frequency of 9.7 × 1018 Hz. Calculate the value of Planck constant from these data.
Wavelength of X-ray = λ = hc/eV ….(1)
We know, γ = c/ λ, using in (1), we get
h = eV/ γ = [40×103]/[9.7×1018] x e = 4.12 x 10-15 eVs
Question 11: An X-ray tube operates at 40 kV. Suppose the electron converts 70% of its energy into a photon at each collision. Find the lowest three wavelengths emitted from the tube. Neglect the energy imparted to the atom with which the electron collides.
Energy utilized by the electron = E = 70% energy into a photon
=> E = 70/100 x 40 x 103 eV = 28 x 103 eV
Wavelength of the X-ray = λ = hc/eV
= 44.35 pm
For the 2nd wavelength (which is 70% of leftover energy)
E = 70/100 x (40-28) x 103 eV = 8.4 x 103 eV
Wavelength of the X-ray = λ = hc/eV
= 148 pm
Similarly, For the 3rd wavelength:
E = 70/100 x (12-8.4) x 103 eV = 25.2 x 102 eV
Wavelength of the X-ray = λ = hc/eV
= 493 pm
Question 12: The wavelength of Kα X-ray of tungsten is 21.3 pm. It takes 11.3 keV to knock out an electron from the L shell of a tungsten atom. What should be the minimum accelerating voltage across an X-ray tube having tungsten target which allows production of Kα X-ray?
Energy required to knock out an electron from L-shell = EL = 11.3 KeV
Wavelength of X-ray = 21.3 pm
Voltage = 11.3 kV
The energy gap between K and L shell,
EK – EL = [1242×10-9]/[21.3×10-12] = 58.309 KeV
Ek = 11.3 + 58.309 = 69.609 KeV
The accelerating voltage across the X-ray tube for the production of Kα X-ray is Vk = 69.609 KeV
Question 13: The Kβ X-ray of argon has a wavelength of 0.36 nm. The minimum energy needed to ionize an argon atom is 16 eV. Find the energy needed to knock out an electron from the K shell of an argon atom
Energy of Kβ x-ray of argon, E = [1242×10-9]/[0.36×10-9] = 3450 eV[Given: Wavelength of Kβ X-ray of argon = 0.36 nm]
Energy needed to knock out an electron from K shell can be calculated as:
Ek = 3450 + 16 eV
Ek = 3.47 KeV (approx.)
Question 14: The Kα X-rays of aluminum (Z = 13) and zinc (Z = 30) have wavelengths 887 pm and 146 pm respectively. Use Moseley’s law √v = a(Z – b) to find the wavelength of the Kα X-ray of iron (Z = 26).
Let γ1 and γ2 are the frequencies of Kα X-rays of aluminum and zinc respectively
So, γ1 = [3×108]/[887 x 10-12] = 33.82 x 1016 Hz
and γ2 = [3×108]/[146 x 10-12] = 2.055 x 1018 Hz
√γ = a(Z – b)
5.815 x 108 = a(13 – b) …(a)
1.4331 x 109 = a(30 – b) …(a)
Dividing above equations, we get
(13-b)/(30-b) = [5.815 x 108 ]/[1.4331 x 109 ] = 0.4057
=> (13-b)/(30-b) = 0.4057
=> (13-b)0.4057 = (30-b)
=> b = 1.3949
(a)=> a = [5.815 x 108]/(13-1.3949) = 5 x 107
Now, let us find the wavelength of the Kα X-ray of iron:
Frequency of the iron is given as √γ = (5 x 107)(26 – 1.39) = 123.05 x 107
or γ = 5.1413 x 1014
We know, γ = c/λ
or λ = c/γ = [3×108]/[5.1413 x 1014] = 198 pm
Question 15: A certain element emits Kα X-ray of energy 3.69 keV. Use the data from the previous problem to identify the element.
Energy of Kα X-rays= 3.69 KeV (Given)
Using formula to find the Wavelength, or λ = hc/eV
=> λ = [1242×10-9]/ = 0.34 x 10-9 m[Because 3.69 KeV = 3690 eV]
Using Moseley’s law, √γ = a(Z – b) ….(1)
Where γ = c/λ
√γ = √c/λ = √[3×108]/√[0.34×10-9] = 9.39 x 108
(1)=> 9.39 x 108 = (5 x 107)(Z – 1.39)[Note: Used values of a and b from previous problem]
=> z = 20.17 = 20 (approx)
Atomic number (Z) = 20
Therefore, element with Atomic number 20 is calcium (Ca).
Question 16: The Kβ X-rays from certain elements are given below. Draw a Moseley-type plot of √v versus Z for Kβ radiation.
Using energy frequency relation to find the value of v.
i.e. Frequency = Energy/h
Where h = Planck’s constant
and energy in KeV.
The required graph is:
Question 17: Use Moseley’s law with b = 1 to find the frequency of the Kα X-ray of La (Z = 57) if the frequency of the Kα X-ray of Cu (Z = 29) is known to be 1.88 × 1016 Hz.
Using Moseley’s law, √ν = a(Z – b) ….(1)
Where ν = c/λ
Given b = 1
Given: ZLa = 57 and ZCu = 29
=> νLa = 1.88 x 1018 ((57-1)/(29-1))2 = 7.52 x 1018
=> νLa = 7.52 x 1018 Hz
Question 18: The Kα and Kβ X-rays of molybdenum have wavelengths 0.71 angstrom and 0.63 angstrom respectively. Find the wavelength of Lα X-ray of molybdenum.
Solution: The Kα and Kβ X-rays of molybdenum have wavelengths 0.71 angstrom and 0.63 angstrom respectively.
Let λ1 = 0.71 angstrom and λ2 = 0.63 angstrom
To find: Wavelength of Lα
Energy of Kα x-rays is given by: Kα = EK – EL
Energy of Kβ is given by: Kβ = EK – EM and
Energy of Lα is given by: Lα = EK – EM
Relating above equations, we have
Lα = Kα – Kβ
= [3×108]/[0.63×10-10] – [3×108]/[0.71×10-10]
or Lα = 0.536 x 1018 Hz
We know, λ = c/v = (3×108)/(0.536×1018) = 5.6 angstrom.
Question 19: The wavelengths of Kα and Lα X-rays of a material are 21.3 pm and 141 pm respectively. Find the wavelength of Kβ X-ray of the material.
The wavelengths of Kα and Lα X-rays of a material are 21.3 pm and 141 pm respectively.
Let E1, E2 and E3 are the energies of Kα, Lα and Kβ.
Using all three relations, we have
E3 = E1 + E2
=> E3 = 1242/[21.3×10-3] + 1242/[141×10-5]
= 58.309 x 103 + 8.8085 x 103
= 67.118 x 103 eV
Now, Find the wavelength of Kβ X-ray of the material:
λ = hc/E = (1242)/(67.118 x 103) = 18.5 pm.
Question 20: The energy of a silver atom with a vacancy in K shell is 25.31 keV, in L shell is 3.56 keV and in M shell is 0.530 keV higher than the energy of the atom with no vacancy. Find the frequency of Kα, Lβ and Lα X-rays of silver.
Let E1, E2 and E3 are the energies of K shell, L shell and M shell.
So, E1 = 25.31 KeV , E2 = 3.56 keV and E3 = 0.530 keV
Let v1, v2 and v3 be the frequencies of Kα, Lβ and Lα x-rays respectively.
Kα x-ray emitted when the transition takes place between l and k shells.
=> Kα = E1 – E2 = h v1
or v1 = (E1 – E2)/h = (25.31-3.56)/(6.63×10-34) x 1.6 x 10-19 x 103
= 5.25 x 1018 Hz
kβ x-ray is emitted when the transition takes place between k and m shells
=> Kβ = E1 – E3 = h v2
or v2 = (E1 – E3)/h = (25.31-0.53)/(6.63×10-34) x 1.6 x 10-19 x 103
= 5.985 x 1018 Hz
Lα x-ray is emitted when the transition takes place between l and m shells
=> Lα = E2 – E3 = h v3
or v3 = (E2 – E3)/h = (3.56-0.53)/(6.63×10-34) x 1.6 x 10-19 x 103
= 7.32 x 1017 Hz