Dual Nature of Radiation and Matter Test

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Dual Nature of Radiation and Matter Test - 46

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Weekly Quiz Competition

Question 1
1 / -0

A cesium photo cell, with a steady potential difference of $$60$$ volt across it, is illuminated by a small bright light placed $$50$$ cm away. When the same light is placed one meter away, the photoelectrons emerging from the photo cell : (assume that potential difference applied is sufficient to produce saturation current)

A
each carry one quarter of their previous energy

B
each carry one quarter of their previous momentum

C
are half as numerous

D
are one quarter as numerous

Solution

number $$\int {photo} $$ electronic $$ \propto {1 \over {{d^2}}}$$

                $$\displaystyle {{{N_1}} \over {{N_2}}} = {\left( {{{{d_2}} \over {{d_1}}}} \right)^2}$$

                $$ = \displaystyle {\left( {{{100} \over {50}}} \right)^2}$$

                $$ = \displaystyle {4 \over 1}$$

                $${N_2} = \displaystyle {{{N_1}} \over 4}$$
Question 2
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A free particle with initial kinetic energy $$E$$ and de-broglie wavelength $$\lambda$$ enters a region in which it has potential energy $$U$$. What is the particle's new de-Broglie wavelength?

A
$$\lambda (1-U/E)^{-1/2}$$

B
$$\lambda (1-U/E)$$

C
$$\lambda (1-U/E)^{-1}$$

D
$$\lambda (1-U/E)^{1/2}$$

Solution
Question 3
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The De-Broglie wavelength associated with electrons revolving round the nucleus in a hydrogen atom in ground state, will be-

A
$$0.3\ A^{\circ}$$

B
$$3.3\ A^{\circ}$$

C
$$6.62\ A^{\circ}$$

D
$$10\ A^{\circ}$$

Solution

According to $$Bohr's$$ postulate $$mvr=\dfrac{nh}{2\pi}$$ so the momentum in ground state$$(n=1)$$ will be $$p=mv=\dfrac{h}{2\pi r}$$
Required wavelength will be $$\lambda= \dfrac{h}{p}=\dfrac{h}{h/2 \pi r}=2\pi r$$
Value of radius for ground state in hydrogen is $$r=0.53Angstrom$$
Using that we get $$\lambda=3.32Angstrom$$
Option B is correct.
Question 4
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The ratio of $$d-$$Broglie wavelength of molecular of hydrogen and oxygen kept in two vessels separately at $$27^oC$$ and $$127^oC$$ respectively is:

A
$$\sqrt{\dfrac{4}{3}}$$

B
$$\sqrt{\dfrac{3}{32}}$$

C
$$\dfrac{8}{\sqrt{3}}$$

D
$$\dfrac{1}{8}$$

Solution

We know that,
$$\dfrac { { \lambda H }_{ 2 } }{ { \lambda H }_{ e } } =\dfrac { \sqrt { { M }_{ { H }_{ e } }{ T }_{ { H }_{ e } } } }{ \sqrt { { M }_{ { H }_{ 2 } }{ T }_{ { H }_{ 2 } } } } $$
Now,
$$\dfrac { { \lambda H }_{ 2 } }{ { \lambda H }_{ e } } =\sqrt { \dfrac { 4\left( 127+273 \right) }{ 2\left( 27+273 \right) } } $$
$$\dfrac { { \lambda H }_{ 2 } }{ { \lambda H }_{ e } } =\sqrt { \dfrac { 1600 }{ 600 } } $$
$$\dfrac { { \lambda H }_{ 2 } }{ { \lambda H }_{ e } } =\sqrt { \dfrac { 8 }{ 3 } } =\sqrt { \dfrac { 4 }{ 3 } } $$
Question 5
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Which of the following figures represent the variation of particle momentum and the associated de-Broglie wavelength?

A

B

C

D

Solution

Hint:- According to de-Broglie hypothesis a wave is associated with a moving particle whose wavelength is given by
$$\lambda =\dfrac {h}{P}$$

Explanation:-
As wavelength is given by
$$\lambda =\dfrac {h}{P}$$
$$\lambda \propto \dfrac{1}{P}$$
$$P \propto \dfrac {1}{\lambda}$$
As momentum is inversely proportional to wavelength of light so graph between momentum and wavelength is hyperbolic.
Question 6
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A body of mass $$m$$ is dropped freely from a height $$h$$. The de-Broglie wavelength of the body as it reaches the ground is:

A
$$\dfrac{h}{\sqrt{gH}}$$

B
$$\dfrac{h}{m\sqrt{gH}}$$

C
$$\dfrac{h}{2\sqrt{gH}}$$

D
$$\dfrac{h}{m\sqrt{2gH}}$$

Solution

We know ,
Velocity$$v = \sqrt {2gH} $$
$$\begin{array}{l} \lambda =\frac { h }{ { mv } } \\ =\frac { h }{ { m\sqrt { 2gH } } } \end{array}$$
$$\therefore$$ Option $$D$$ is correct.
Question 7
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what is the angular momentum of an electron of de$$-$$broglie wavelength $$\lambda ?$$ given $$r$$ is the radius of orbit$$.$$

A
$$\dfrac{{rh}}{\lambda }$$

B
$$\dfrac{{2rh}}{\lambda }$$

C
$$\dfrac{{3rh}}{\lambda }$$

D
$$\dfrac{{4rh}}{\lambda }$$

Solution

$$mvr = \dfrac{h}{\lambda }$$
$$r = \dfrac{{rh}}{\lambda }$$
Hence,
option $$(A)$$ is correct answer.
Question 8
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According to de Broglie, wavelength of electron in second orbit is $$10^{-9}$$ meter. Then the circumference of orbit is :-

A
$$10^{-9}$$

B
$$2 \times 10^{-9}$$

C
$$3 \times 10^{-9}$$

D
$$4 \times 10^{-9}$$

Solution
Question 9
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A particle is dropped from a height H. The de-Broglie wavelength of the particle as a function of height is proportional to

A
$$H$$

B
$$H\dfrac{1}{2}$$

C
$$H^o$$

D
$$H\dfrac{-1}{2}$$

Solution

Solution :
Velocity $$v= \sqrt{2gH}$$
$$\lambda =\dfrac{h}{mv}$$
$$= \dfrac{h}{m\sqrt{2gh}}$$
$$\lambda \alpha H^{-1/2}$$
Question 10
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Stopping potential $$(V_0)$$ frequency $$(v)$$ of radiation graph for three different emitter surfaces $$A, B$$ and $$C$$ as shown in figure, then
(a) threshold frequency of $$A < B <C$$
(b) work function of $$A < B <C$$
(c) work function of $$A = B = C$$
(d) threshold wavelength of $$A < B <C$$
(e) ratio of the slopes of $$A , B, C$$ id $$1 : 1 : 1$$

A
$$a, b, d, e$$ are correct

B
$$a, b, d$$ are correct

C
$$c, d, e$$ are correct

D
$$a, c, d, e$$ are correct