Q: What is the speed of a photon with respect to another photon if(a) the two photons are going in the same direction and (b) they are going in opposite directions?

(a) In relativity, the relative speed of two objects (vrel ) moving in the same direction with speeds u and v is given byvrel=1−c2uvu−v…(1) As the photons are moving with the speed of light, u=c and v=c. On substituting the values of u and v in equation (1), we get:vrel =0 Thus, relative velocity of a photon with respect to another photon will be 0 , when they are going in the same direction.(b) In relativity, relative speed of two objects moving in opposite directions with speeds u and v is given byvrel=1+c2uvu+v…(2) We know that a photon travels with the speed of light. Therefore, u=c and v=c On substituting the values of u and v in equation (2), we get:vrel=c Thus, the relative velocity of a photon with respect to another photon will be equal to the speed of light when they are going in opposite directions.

Q: The threshold wavelength of a metal is λ0. Light of wavelength slightly less than λ0 is incident on an insulated plate made of this metal. It is found that photoelectrons are emitted for sometime and after that the emission stops. Explain.

When light of wavelength less than λ0 is incident on the metal surface, the free electrons of the metal will gain energy and come out of the metal surface. As the metal plate is insulated (it is not connected with the battery), the free electrons of the metal will not be replaced by the other electrons. Hence, photoelectron emission will stop after some time.

Question 1

Can we find the mass of a photon by the definition p = mv?

Accepted Answer

No, we cannot find the mass of a photon by the definition p=mv . The equation p=mv is valid only for objects that move with a velocity that is much slower than the speed of light. The momentum of a relativistic particle like photon is given by pc=E2−m2c4​.A photon has zero rest mass. Therefore, on putting m=0 in the equation, we get p=cE​ , which is the valid equation for a photon.

Question 2

Is it always true that for two sources of equal intensity, the number of photons emitted in a given time are equal?

Accepted Answer

Let the source's area be A, and intensity of the source be I. The energy of each emitted photonis E. Then, the number of photons emitted in a given time will be n=I/(AE) If the areas of the sources and the wavelengths of light emitted by the two sources are different, then the number of photons emitted will be different.

Question 3

What is the speed of a photon with respect to another photon if(a) the two photons are going in the same direction and (b) they are going in opposite directions?

Accepted Answer

(a) In relativity, the relative speed of two objects (vrel ​) moving in the same direction with speeds u and v  is given byvrel​=1−c2uv​u−v​…(1) As the photons are moving with the speed of light, u=c and v=c. On substituting the values of u  and v in equation (1), we get:vrel ​=0 Thus, relative velocity of a photon with respect to another photon will be 0 , when they are going in the same direction.(b) In relativity, relative speed of two objects moving in opposite directions with speeds u and v is given byvrel​=1+c2uv​u+v​…(2) We know that a photon travels with the speed of light. Therefore, u=c and v=c On substituting the values of u and v in equation (2), we get:vrel​=c Thus, the relative velocity of a photon with respect to another photon will be equal to the speed of light when they are going in opposite directions.

Question 4

Can a photon be deflected by an electric field? By a magnetic field?

Accepted Answer

Photons are electrically neutral. Hence, they are not deflected by electric and magnetic fields.

Question 5

A hot body is placed in a closed room maintained at a lower temperature. Is the number of photons in the room increasing?

Accepted Answer

As the hot body is placed in a closed room maintained at a lower temperature, there will be transfer of heat in the room through convection and radiation. Heat radiation also consists of photons; therefore, photons will be emitted by the hot body. Hence, the number of photons in the room will increase.

Question 6

Should the energy of a photon be called its kinetic energy or its internal energy?

Accepted Answer

Relativistic equation of energy:E2=p2c2+m2c4…(1) Here, p2c2= kinetic energy of photonm02​c4= internal energy of photonWe know photons have zero rest mass. Therefore, m0​=0 Substituting the value of m0​=0 in equation (1), we get:E=pc Thus, the energy of a photon should be called its kinetic energy.

Question 7

In an experiment on photoelectric effect, a photon is incident on an electron from one direction and the photoelectron is emitted almost in the opposite direction. Does this violate the principle of conservation of momentum?

Accepted Answer

No, it does not violate the principle of conservation of momentum. In the photon-electron collision, the energy and momentum are conserved.

Question 8

It is found that yellow light does not eject photoelectrons from a metal. Is it advisable to try with orange light or with green light?

Accepted Answer

Photoelectrons are emitted from a metal's surface if the frequency of incident radiation is more than the threshold frequency of the given metal surface. As yellow light does not eject photoelectrons from a metal it means that the threshold frequency of the metal is more than the frequency of yellow light. Since the frequency of orange light is less than the frequency of yellow light, therefore it will not be able to eject photoelectrons from the metal's surface. The frequency of green light is more than the frequency of yellow light. Hence, when it is incident on the metal surface, it will eject electrons from the metal.

Question 9

It is found that photosynthesis starts in certain plants when exposed to sunlight, but it does not start if the plants are exposed only to infrared light. Explain.

Accepted Answer

Photosynthesis starts when a plant is exposed to visible light. The visible light's photons possess just enough energy to excite the electrons of molecules of the plant without causing damage to its cells. Infrared rays have less frequency than visible light. Due to this, the energy of the photons of infrared rays are not sufficient to initiate photosynthesis. Therefore, photosynthesis does not start if plants are exposed only to infrared light.

Question 10

The threshold wavelength of a metal is λ0​. Light of wavelength slightly less than λ0​ is incident on an insulated plate made of this metal. It is found that photoelectrons are emitted for sometime and after that the emission stops. Explain.

Accepted Answer

When light of wavelength less than λ0​ is incident on the metal surface, the free electrons of the metal will gain energy and come out of the metal surface. As the metal plate is insulated (it is not connected with the battery), the free electrons of the metal will not be replaced by the other electrons. Hence, photoelectron emission will stop after some time.

Question 11

Is p=E/c valid for electrons?

Accepted Answer

From Einstein's mass- energy equation,​E=mc2⇒E=1−c2v2​​m0​c2​​ Relativistic momentum,p=mv⇒p=1−c2v2​​m0​c2​​ Combining the above equations, we get:E2=m02​c4+p2c2 From the above equation, it is clear that for p=E/c to be valid, the rest mass of the body should be zero. As electrons do not have zero rest mass, this equation is not valid for electrons.

Question 12

Consider the de-Broglie wavelength of an electron and a proton. Which wavelength is smaller if the two particles have(a) the same speed (b) the same momentum (c) the same energy?

Accepted Answer

de-Broglie wavelength,λ=mvh​ where h= Planck's constantm=mass of the particlev=velocity of the particle(a) It is given that the speed of an electron and proton are equal.It is clear from the above equation thatλ∝m1​ As mass of a proton, mp​>mass of an electron, me​, the proton will have a smaller wavelength compared to the electron.(b)λ=ph​(∵p=mv)So, when the proton and the electron have same momentum, they will have the same wavelength.(c) de-Broglie wavelength (λ) is also given byλ=2mE​h​ where E=energy of the particle.Let the energy of the proton and electron be E . Wavelength of the proton,λp​=2mp​E​h​…(1) Wavelength of the electron,λe​=2me​E​h​…(2 Dividing (2) by (1), we get:​λp​λe​​=mp​​me​​​⇒λp​λe​​<1⇒λe​<λp​​ It is clear that the proton will have smaller wavelength compared to the electron.

Question 13

If an electron has a wavelength, does it also have a colour?

Accepted Answer

Colour is a characteristic of electromagnetic waves. Electrons behave as a de-Broglie wave because of their velocity. A de-Broglie wave is not an electromagnetic wave and is one dimensional. Hence, no colour is shown by an electron.

Question 14

Planck's constant has the same dimensions as:

Accepted Answer

Force ×distance timePlanck's constant,h=vE​= frequency  Force × distance ​ ⇒h=force ×distance time

Question 15

Two photons having:

Accepted Answer

Two photons having equal linear momenta have equal wavelengths is correct. As in the rest of the options magnitude of momentum or energy can be same because energy and momentum are inversely proportional to wavelength. But the direction of propagation of the photons can be different.Hence the correct option is D.

Question 16

Let p and E denote the linear momentum and energy of a photon. If the wavelength is decreased:

Accepted Answer

From the de-Broglie relation, wavelength,λ=ph​…(1)⇒p=λh​​ Here, h= Planck's constantp=momentum of electronIt is clear from the above equation that p∝λ1​.Thus, if the wavelength (λ) is decreased, then momentum (p)will be increase. Relation between momentum and energy:p=2mE​ Here, E= energy of electronm=mass of electronSubstituting the value of p in equation (1), we get:​λ=2mE​h​⇒E​=λ2m​h​⇒E=2mλ2h2​​ Thus, on decreasing λ , the energy will increase.

Question 17

Let nr​ and nb​ be respectively the number of photons emitted by a red bulb and a blue bulb of equal power in a given time.

Accepted Answer

The two bulbs are of equal power. It means that they consume equal amount of energy per unit time.Now, as the frequency of blue light (fb​) is higher than the frequency of red light (fr​),hfb​>hfr​. Hence, the energy of a photon of blue light is more than the energy of a photon of red light. Thus, a photon of blue light requires more energy than a photon of red light to be emitted. For the same energy given to the bulbs in a certain time, the number of photons of blue light will be less than that of red light.∴nr​>nb​ (As the amount of energy emitted from the two bulb is same)

Question 18

The equation E = pc is valid:

Accepted Answer

The equation E=pc is valid for a particle with zero rest mass. The rest mass of a photon is zero, but the rest mass of an electron is not zero. So, the equation will be valid for photon, and not electron.

Question 19

The work function of a metal is hv0−​. Light of frequency v  falls on this metal. The photoelectric effect will take place only if:

Accepted Answer

As the work function of the metal is hv0​, the threshold frequency of the metal is v0​.For photoelectric effect to occur, the frequency of the incident light should be greater than or equal to the threshlod frequency of the metal on which light is incident.

Question 20

Light of wavelength λ falls on a metal having work function hc/λ0​. Photoelectric effect will take place only if:

Accepted Answer

As the work-function of the metal is hc/λ0​, its threshold wavelength is λ0​.For photoelectric effect, the wavelength of the incident light should be less than or equal to the threshold wavelength of the metal on which light is incident.

Chapter 20 : Photoelectric Effect and Wave Particle Duality