Dual Nature of Radiation and Matter Test

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Dual Nature of Radiation and Matter Test - 48

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Question 1
1 / -0

A particle of mass M at rest decays into particles of masses $$m_{1} \ and \ m_{2}$$ having non -zero velocities. The ratio of the de broglie wavelengths of the particles $$\lambda _{1}/\lambda _{2}$$ is :

A
$$m _{1}/m _{2}$$

B
$$m _{2}/m _{1}$$

C
1

D
m1.m2

Solution

When mass $$M$$ at rest (initial momentum$$=0$$) decays to smaller masses $${m}_{1}$$ and $${m}_{2}$$, energies have same momentum (From conservation of momentum)
$${ \vec { P } }_{ 1 }+{ \vec { P } }_{ 2 }=0\Rightarrow { P }_{ 1 }={ P }_{ 2 }$$ (magnitude)
And de-Broglie wavelength
$${ \lambda }_{ 1 }=\cfrac { h }{ { P }_{ 1 } } ;{ \lambda }_{ 2 }=\cfrac { h }{ { P }_{ 2 } } $$ (H-Plancks constant)
Thus we get
$${ \lambda }_{ 1 }={ \lambda }_{ 2 }\Rightarrow \cfrac { { \lambda }_{ 1 } }{ { \lambda }_{ 2 } } =1$$
Question 2
1 / -0

De-Broglie wavelength associated with thermal neutrons at room temperature of $$27^{0}$$C is :

A
$$1.452\times 10^{-10}$$m

B
$$2.57\times 10^{-10}$$

C
$$0.452\times 10^{-10}$$m

D
$$3.452\times 10^{-10}$$m

Solution

Given,

$$T=27+273=300K$$

$${ \text { Boltzmann's constant. } }$$

$${ k = 1.38 \times 10 ^ { - 23 } J m o l ^ { - 1 } k ^ { - 1 } }$$

Kinetic energy of neutron at absolute temperature

$$\text{ T is given by, }E=\dfrac{3}{2}kT$$

$${ \lambda = \dfrac { h } { \sqrt { 2 m E } } }$$$$=\dfrac{h}{\sqrt{3mkT}}$$

$${ \lambda = \dfrac { 6.63 \times 10 ^ { - 34 } } { \sqrt { 3 \times 1.675 \times 10 ^ { - 27 } \times 1.38 \times 10 ^ { - 23 } \times 300 } } }$$$$=1.45\times {{10}^{-10}}m\text{ }$$

Hence, De Broglie wavelength is $$1.45\times {{10}^{-10}}m\text{ }$$
Question 3
1 / -0

The ratio of wavelength of deutron and proton accelerated through the same potential difference will be-

A
$$\dfrac { 1 }{ \sqrt { 2 } }$$

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

C
$$\dfrac { 1 }{ 2 }$$

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

Solution
Question 4
1 / -0

The minimum frequency v of continuous X-rays is related to the applied potential difference V as?

A
$$v\propto \sqrt{V}$$

B
$$v\propto V$$

C
$$v\propto V^{3/2}$$

D
$$v\propto V^2$$

Solution

As we know,
$$E=hv=eV $$
$$\Rightarrow v\propto V$$
Question 5
1 / -0

If the mass of neutron $$= 1.7 \times 10 ^ { - 27} \mathrm { kg }$$ . Then the de broglie wavelength of neutron of energy3$$\mathrm { eV }$$ is

A
$$1.6 \times 10 ^ { - 10 } \mathrm { m }$$

B
$$1.6 \times 10 ^ { - 11 } \mathrm { m }$$

C
$$1.4 \times 10 ^ { - 10 } \mathrm { m }$$

D
$$1.4 \times 10 ^ { - 11 } \mathrm { m }$$

Solution
Question 6
1 / -0

If Bohr radius is $$r_{0},$$ the corresponding de Broglie wavelength of the electron is

A
$$\left ( \dfrac{2\pi }{r_{o}} \right )$$

B
$$\left ( \dfrac{r_{o} }{2\pi } \right )$$

C
$$\left ( \dfrac{1}{2\pi r_{o}} \right )$$

D
$${2\pi r_{o}} $$

Solution

Given that,

Radius $$r={{r}_{0}}$$

We know that,

$$2\pi r=n\lambda $$

Suppose $$n=1$$

Then,

$$\lambda =2\pi {{r}_{0}}$$

Hence, this is the required solution
Question 7
1 / -0

A microscope using suitable photons is employed to locate an electron in an atom within a distance of $$0.1 \ \mathring {A}$$. The uncertainty of its velocity is

A
$$5.79 \times 10^5 \ ms^{-1}$$

B
$$8.79 \times 10^5 \ ms^{-1}$$

C
$$5.79 \times 10^4 \ ms^{-1}$$

D
$$3.79 \times 10^5 \ ms^{-1}$$

Solution

According to Heisenberg's Uncertainty principle,
$$m=9.11\times { 10 }^{ -31 }kg\quad \quad h=6.626\times { 10 }^{ -34 }Js$$
$$\Delta v=\dfrac { h }{ 4\pi m\Delta x } $$
$$\Delta v=\dfrac { 6.626\times { 10 }^{ -34 } }{ 4\times 3.14\times 9.11\times { 10 }^{ -31 }\times 0.1\times { 10 }^{ -10 } } $$
$$\Delta v=5.79\times { 10 }^{ 6 }m/s$$
Question 8
1 / -0

What is the maximum wavelength of the Lyman series of $$He^+$$ ion ?

A
$$\frac{1}{3R}$$

B
$$3R$$

C
$$\frac{4}{4R}$$

D
$$None of these$$
Question 9
1 / -0

Photons of wavelength $$6620A^0 $$ are incident normally on a perfectly reflecting screen. Calculate the number of photons paper per second falling on the screen as the total power of photons such that the exerted force is 1N.

A
$$5\times 10^{26}$$

B
$$5\times 10^{25}$$

C
$$1.5\times 10^{8}$$

D
None of these

Solution

Momentum, $$p=\dfrac { h }{ \lambda } $$ reflected normally
$$\Delta p=\dfrac { 2h }{ \lambda } $$
Force $$=$$ rate of charge of momentum
$$1=\frac { \Delta p }{ t } $$ for each photon $$(t=1sec)$$
let $$n$$ be no. of photon
$$1=\dfrac { n\times 2\left( 6.63\times { 10 }^{ -34 } \right) }{ 6620\times { 10 }^{ -10 } } $$
$$n=\dfrac { 6620\times { 10 }^{ -10 } }{ 2\times 6.63\times { 10 }^{ -34 } } $$
$$\approx 5\times { 10 }^{ 26 }$$ photons.
Question 10
1 / -0

When an electron de-excited back from $$(n+1)^th$$ state to $$n^th$$ state in a hydrogen like atoms, wavelength of radiation emitted is $$\lambda_{1}(n>>1)$$. In the same atom de-broglies wavelength associated with an electron in $$n^th$$ state is $$\lambda_{2}$$. Then $$\dfrac{\lambda_{1}}{\lambda_{2}}$$ is proportional to

A
$$\dfrac{1}{n}$$

B
$$n$$

C
$$n^{2}$$

D
$$n^{3}$$

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Dual Nature of Radiation and Matter Test - 48

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Dual Nature of Radiation and Matter Test - 48

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Question 1

$$m _{1}/m _{2}$$

$$m _{2}/m _{1}$$

1

m1.m2

Solution

Question 2

$$1.452\times 10^{-10}$$m

$$2.57\times 10^{-10}$$

$$0.452\times 10^{-10}$$m

$$3.452\times 10^{-10}$$m

Solution

Question 3

$$\dfrac { 1 }{ \sqrt { 2 } }$$

$$\sqrt { \dfrac { 2 }{ 1 } }$$

$$\dfrac { 1 }{ 2 }$$

$$\dfrac { 2 }{ 1 }$$

Solution

Question 4

$$v\propto \sqrt{V}$$

$$v\propto V$$

$$v\propto V^{3/2}$$

$$v\propto V^2$$

Solution

Question 5

$$1.6 \times 10 ^ { - 10 } \mathrm { m }$$

$$1.6 \times 10 ^ { - 11 } \mathrm { m }$$

$$1.4 \times 10 ^ { - 10 } \mathrm { m }$$

$$1.4 \times 10 ^ { - 11 } \mathrm { m }$$

Solution

Question 6

$$\left ( \dfrac{2\pi }{r_{o}} \right )$$

$$\left ( \dfrac{r_{o} }{2\pi } \right )$$

$$\left ( \dfrac{1}{2\pi r_{o}} \right )$$

$${2\pi r_{o}} $$

Solution

Question 7

$$5.79 \times 10^5 \ ms^{-1}$$

$$8.79 \times 10^5 \ ms^{-1}$$

$$5.79 \times 10^4 \ ms^{-1}$$

$$3.79 \times 10^5 \ ms^{-1}$$

Solution

Question 8

$$\frac{1}{3R}$$

$$3R$$

$$\frac{4}{4R}$$

$$None of these$$

Question 9

$$5\times 10^{26}$$

$$5\times 10^{25}$$

$$1.5\times 10^{8}$$

None of these

Solution

Question 10

$$\dfrac{1}{n}$$

$$n$$

$$n^{2}$$

$$n^{3}$$

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