A particle A with a mass mA is moving with
Question: A particle A with a mass mA is moving with a velocity v and hits a particle B (mass mB) at rest (one dimensional motion). Find the change in the de Broglie wavelength of the particle A. Treat the collision as elastic. Solution: According to the law of conservation of momentum, mAv + mB0 = mAv1+ mBv2 mA(v-v1) = mBv2 1/2 mAv2= 1/2 mAv12+ 1/2mBv22 mA(v-v1)(v+v1) = mBv22 v1 = (mA-mB/mA+mB)v v2 = (2mA/mA+mB)v initial = h/mAv final = h/mAv1 ∆ = final initial = h/mAv(mA+mB/mA-mB)-1...
Read More →A student performs an experiment on the photoelectric effect,
Question: A student performs an experiment on the photoelectric effect, using two materials A and B. A plot of Vstop vs is given in the figure. (i) Which material A or B has a higher work function? (ii) Given the electric charge of an electron = 1.6 1019 C, find the value of h obtained from the experiment for both A and B. Comment on whether it is consistent with Einsteins theory Solution: (i) The threshold frequency of A is vOA= 5 1014Hz The threshold frequency of B is vOB= 10 1014Hz Work funct...
Read More →Consider an electron in front of the metallic surface
Question: Consider an electron in front of the metallic surface at a distance d (treated as an infinite plane surface). Assume the force of attraction by the plate is given as 2 2 0 1 4 4 q d Calculate work in taking the charge to an infinite distance from the plate. Taking d = 0.1nm, find the work done in electron volts. [Such a force law is not valid for d 0.1nm]. Solution: The figure shows that an electron is displaced slowly. The distance covered by the electron is x and this happens by the ...
Read More →In a single throw of two dice, determine the probability of not getting
Question: In a single throw of two dice, determine the probability of not getting the same number on the two dice. Solution: We know that Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total outcomes are $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)$, $(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)$, $(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)$, $(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)$, $(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)$, $(6,1),(6,2),(6,3),(6,4),(6,5...
Read More →Consider a thin target (10–2m square, 10–3m thickness) of sodium,
Question: Consider a thin target (102m square, 103m thickness) of sodium, which produces a photocurrent of 100A when a light of intensity 100W/m2 ( = 660nm) falls on it. Find the probability that a photoelectron is produced when a photon strikes a sodium atom. [Take density of Na = 0.97 kg/m3]. Solution: Area = 10-4m2 Thickness = 10-3m Current = 10-4A Intensity = 100 W/m2 Mass = (volume)(density) = 0.97 10-4gm No.of target atoms = 0.254 1019 Total energy = nhv Where n = 3.3 1016 The probability ...
Read More →A neutron beam of energy E scatters from atoms
Question: A neutron beam of energy E scatters from atoms on a surface with a spacing d = 0.1nm. The first maximum of intensity in the reflected beam occurs at = 30. What is the kinetic energy E of the beam in eV? Solution: From Braggs law, 2d sin = n p = h/ = 6.6 10-24kg m/s E = 1/2 mv2= p2/2m E = 0.085 eV...
Read More →Two particles A and B of de Broglie wavelengths
Question: Two particles A and B of de Broglie wavelengths 1 and 2 combine to form a particle C. The process conserves momentum. Find the de Broglie wavelength of the particle C. (The motion is one dimensional). Solution: According to de-Broglie wavelengths, =h/p p=h/ p1 = h/ 1 p2 = h/ 2 p3 = h/ 3 There are 4 possible cases and they are: Case 1: When p1 and p2 are positive, then 3 = 1 1/ 1+ 2 Case 2: When p1 and p2 are negative, then 3 = 1 2/ 1+ 2 Case 3: When p1 0 and p2 0, then 3 = 1 2/ 2 1 Cas...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(x-1)^{2}}{x^{2}+2 x+2} d x$ Solution: Given $I=\int \frac{(x-1)^{2}}{x^{2}+2 x+2} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ $\Rightarrow \int \frac{(x-1)^{2}}{x^{2}+2 x+2} d x=\int\left(\frac{-4 x-1}{x^{2}+2 x+2}+1\right) d x$ $=-\int \frac...
Read More →Three unbiased coins are tossed once. Find the probability of getting
Question: Three unbiased coins are tossed once. Find the probability of getting at most 2 tails or at least 2 heads Solution: We know that Probability of occurrence of an event $=\frac{\text { Total no. of Desired outcomes }}{\text { Total no.of outcomes }}$ Let $\mathrm{T}$ be tails and $\mathrm{H}$ be heads Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH Desired outcomes are at least two heads or at most two tails. So, desired outputs are TTH, THT, HTT, THH, HTH, HHT, HHH Tota...
Read More →Two monochromatic beams A and B of equal intensity I,
Question: Two monochromatic beams A and B of equal intensity I, hit a screen. The number of photons hitting the screen by beam A is twice that by beam B. Then what inference can you make about their frequencies? Solution: Let nA be the no.of photons falling per second from beam A and nB be the no.of photons falling per second from beam B. Given, No.of photons hitting the screen by A is twice that of B Therefore, nA = 2nB Energy from the beam A = hvA Energy from the beam B = hvB Intensity = I = n...
Read More →Three unbiased coins are tossed once. Find the probability of getting
Question: Three unbiased coins are tossed once. Find the probability of getting at least 2 tails Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Let $\mathrm{T}$ be tails and $\mathrm{H}$ be heads Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH Desired outcomes are at least two tails. So, the desired outputs are TTH, THT, HTT, TTT Total no.of outcomes are 8 and desired outcomes are 4 T...
Read More →Assuming an electron is confined to a 1nm wide region,
Question: Assuming an electron is confined to a 1nm wide region, find the uncertainty in momentum using the Heisenberg Uncertainty principle. You can assume the uncertainty in position ∆x as 1nm. Assuming p = ∆p, find the energy of the electron in electron volts. Solution: As the electrons rotate in a circular path, ∆r = 1 nm = 10-9m ∆p = h/∆x ∆p = (331/314) 10-25 E = 1/2 mv2= ∆p2/2m E = 3.8 10-2eV...
Read More →Consider a metal exposed to light of wavelength 600 nm.
Question: Consider a metal exposed to light of wavelength 600 nm. The maximum energy of the electron doubles when light of wavelength 400 nm is used. Find the work function in eV. Solution: Let K1 and K2 be the maximum energy emitted by the electrons when 600 and 400 nm wavelength is used. K2 = 2K1 Therefore, the work function, ϕ = 1.03 eV...
Read More →Consider the figure given for photoemission.
Question: Consider the figure given for photoemission. How would you reconcile with momentum-conservation? Note light (photons) have momentum in a different direction than the emitted electrons. Solution: When the photons strike the metal surface, there is a transfer of momentum to the atoms on the metal surface which results in a decrease in speed of the photons. The momentum of photons is transferred to the nucleus and electrons on the metal surface. The electrons emitted are in the opposite d...
Read More →Three unbiased coins are tossed once.
Question: Three unbiased coins are tossed once. Find the probability of getting at most 2 tails Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Let $\mathrm{T}$ be tails and $\mathrm{H}$ be heads Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH Desired outcomes are at most two tails. So, desired outputs are THH, HTH, HHT, TTH, THT, HTT, HHH Total no. of outcomes are 8 and desired outcom...
Read More →Three unbiased coins are tossed once.
Question: Three unbiased coins are tossed once. Find the probability of getting exactly one tail Solution: We know that Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Let $\mathrm{T}$ be tails and $\mathrm{H}$ be heads Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH Desired outcomes are exactly one tail. So, desired outputs are THH, HTH, HHT Total no. of outcomes are 8 and desired outcomes are 3 Therefore, t...
Read More →There are two sources of light,
Question: There are two sources of light,each emitting with a power of 100 W. One emits X-rays of wavelength 1nm and the other visible light at 500 nm. Find the ratio of a number of photons of X-rays to the photons of visible light of the given wavelength? Solution: Ex= hvx Ev= hvv Let nxand nvbe the no.of photons from x-rays and visible light are of equal energies and they emit 100 W power. nxhvx= nvhvv nx/nv= vv/vx= x/v nx/nv= 1 nm/500 nm nx:nv= 1:500...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2}+x+1}{x^{2}-x+1} d x$ Solution: Given $I=\int \frac{x^{2}+x+1}{x^{2}-x+1} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ $\Rightarrow \int \frac{x^{2}+x+1}{x^{2}-x+1} d x=\int\left(\frac{2 x}{x^{2}-x+1}+1\right) d x$ $=2 \int\left(\frac{x}{x...
Read More →Three unbiased coins are tossed once. Find the probability of getting
Question: Three unbiased coins are tossed once. Find the probability of getting exactly 2 tails Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Let T be tails and H be heads Total possible outcomes = TTT, TTH, THT, HTT, THH, HTH, HHT, HHH Desired outcomes are exactly two tails. So, desired outputs are TTH, THT, HTT Total no.of outcomes are 8 and desired outcomes are 3 Therefore, the probability of gett...
Read More →Do all the electrons that absorb
Question: Do all the electrons that absorb a photon come out as photoelectrons? Solution: No, not all electrons that are absorbed as a photon come out as a photoelectron....
Read More →There are materials which absorb photons
Question: There are materials which absorb photons of shorter wavelength and emit photons of longer wavelength. Can there be stable substances which absorb photons of larger wavelength and emit light of shorter wavelength? Solution: The wavelength of the photon increases when the frequency decreases. There are two possible cases and they are; case one in which the photons have a smaller wavelength and the energy is consumed against work function. In case two, the photons might have a longer wave...
Read More →If there are two children in a family, find the probability that there is at least one boy in the family
Question: If there are two children in a family, find the probability that there is at least one boy in the family Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no. of Desired outcomes }}{\text { Total no. of outcomes }}$ Let $B$ be Boy and $G$ be Girl Total possible outcomes are BB, BG, GB, GG Our desired outcome is at least one boy. So, BB, BG, GB are desired outputs. Total no. of outcomes are 4 , and the desired outcomes are 3 Therefore, the probability o...
Read More →(i) In the explanation of the photoelectric effect,
Question: (i) In the explanation of the photoelectric effect, we assume one photon of frequency collides with an electron and transfers its energy. This leads to the equation for the maximum energy Emax of the emitted electron as Emax = h 0where 0is the work function of the metal. If an electron absorbs 2 photons (each of frequency ) what will be the maximum energy for the emitted electron? (ii) Why is this fact (two-photon absorption) not taken into consideration in our discussion of the stoppi...
Read More →A proton and an α-particle are accelerated,
Question: A proton and an -particle are accelerated, using the same potential difference. How are the de Broglie wavelengths p and a related to each other? Solution: The proton and -particle are accelerated at the same potential difference. = h/2mqv is proportional to 1/mq p/ a = maqa/mpqp = 8 Therefore, the wavelength of the proton is 8 times the wavelength of -particle....
Read More →In a lottery, there are 10 prizes and 25 blanks. Find the probability of getting a prize.
Question: In a lottery, there are 10 prizes and 25 blanks. Find the probability of getting a prize. Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total no. of outcomes $=10+25=35$ Desired outcomes are prizes. Total no.of desired outcomes $=10$ Therefore, the probability of getting a prize $=\frac{10}{35}$ $=\frac{2}{7}$ Conclusion: Probability of getting a prize is $\frac{2}{7}$...
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