1
MCQ (Single Correct Answer)

JEE Main 2017 (Online) 8th April Morning Slot

The maximum velocity of the photoelectrons emitted from the surface is v when light of frequency n falls on a metal surface. If the incident frequency is increased to 3n, the maximum velocity of the ejected photoelectrons will be :
A
less than $$\sqrt 3 $$ v
B
v
C
more than $$\sqrt 3 \,v$$
D
equal to $$\sqrt 3 \,v$$
2
MCQ (Single Correct Answer)

JEE Main 2017 (Online) 9th April Morning Slot

A Laser light of wavelength 660 nm is used to weld Retina detachment. If a Laser pulse of width 60 ms and power 0.5 kW is used the approximate number of photons in the pulse are :

[Take Planck's constant h $$=$$ 6.62 $$ \times $$ 10$$-$$34 Js]
A
1020
B
1018
C
1022
D
1019
3
MCQ (Single Correct Answer)

JEE Main 2018 (Online) 15th April Morning Slot

Two electrons are moving with non-relativistic speed perpendicular to each other. If corresponding de Broglie wavelength are $${\lambda _1}$$ and $${\lambda _2},$$ their de Broglie wavelength in the frame of reference attached to their center of massis :
A
$${\lambda _{CM}} = {\lambda _1} = {\lambda _2}$$
B
$${\lambda _{CM}} = {{2{\lambda _1}{\lambda _2}} \over {\sqrt {\lambda _1^2 + \lambda _2^2} }}$$
C
$${1 \over {{\lambda _{CM}}}} = {1 \over {{\lambda _1}}} + {1 \over {{\lambda _2}}}$$
D
$${\lambda _{CM}} = \left( {{{{\lambda _1} + {\lambda _2}} \over 2}} \right)$$

Explanation

As we know,

momentum (p) = $${h \over \lambda }$$

Let one perticle is moving in x direction and other is in y dirrection.

$$\therefore\,\,\,\,$$ momentum of each electrons $${h \over {{\lambda _1}}}\widehat i$$ and $${h \over {{\lambda _2}}}\widehat j$$

$$\therefore\,\,\,\,$$ Velocity of each electrons $${h \over {m{\lambda _1}}}\widehat i$$ and $${h \over {m{\lambda _2}}}\widehat j$$

$$\therefore\,\,\,\,$$ Velocity of center of mass $$\left( {{{\overrightarrow \upsilon }_{_{CM}}}} \right)$$ = $${{\upsilon _1^{\widehat i} + \upsilon _2^{\widehat j}} \over 2}$$

Now, velocity of first electron about center of mass,

$${\overrightarrow \upsilon _{_{1CM}}}$$ = $$\upsilon _1^{\widehat i}$$ $$-$$ $$\left( {{{\upsilon _1^{\widehat i} + \upsilon _2^{\widehat j}} \over 2}} \right)$$

= $${{\upsilon _1^{\widehat i} - \upsilon _2^{\widehat j}} \over 2}$$

Similarly,

$${\overrightarrow \upsilon _{_{2CM}}} = {{\upsilon _2^{\widehat j} - \upsilon _1^{\widehat i}} \over 2}$$

Here,

$$\left| {{{\overrightarrow \upsilon }_{_{1CM}}}} \right| = \left| {{{\overrightarrow \upsilon }_{_{2CM}}}} \right| = {1 \over 2}$$

$$\therefore\,\,\,\,$$ $$\upsilon $$ = $${1 \over 2}$$ $$\sqrt {{{{h^2}} \over {{m^2}\lambda _1^2}} + {{{h^2}} \over {{m^2}\lambda _2^2}}} $$

$$ \Rightarrow $$$$\,\,\,\,$$ m $$\upsilon $$ = $${1 \over 2}$$ $${\sqrt {{{{h^2}} \over {\lambda _1^2}} + {{{h^2}} \over {\lambda _2^2}}} }$$

$$ \Rightarrow $$$$\,\,\,\,$$ $${h \over {{\lambda _{CM}}}}$$ = h$${\sqrt {{1 \over {4\lambda _1^2}} + {1 \over {4\lambda _2^2}}} }$$

$$ \Rightarrow $$$$\,\,\,\,$$ $${1 \over {{\lambda _{CM}}}} = {{\sqrt {\lambda _1^2 + \lambda _2^2} } \over {2{\lambda _1}{\lambda _2}}}$$

$$ \Rightarrow $$$$\,\,\,\,$$ $${\lambda _{CM}} = {{2{\lambda _1}{\lambda _2}} \over {\sqrt {\lambda _1^2 + \lambda _2^2} }}$$
4
MCQ (Single Correct Answer)

JEE Main 2018 (Online) 15th April Evening Slot

If the de Broglie wavelengths associated with a proton and an $$\alpha $$-particle are equal, then the ratio of velocities of the proton and the $$\alpha $$-particle will be :
A
4 : 1
B
2 : 1
C
1 : 2
D
1 : 4

Explanation

We know, $${\lambda _p} = {h \over {{p_p}}}$$ = $$ {h \over {{m_p}{v_p}}}$$

Similarly, $${\lambda _\alpha } = {h \over {{m_\alpha }{v_\alpha }}}$$

Given, $${\lambda _p} = {\lambda _\alpha }$$

$$ \Rightarrow $$ $${h \over {{m_p}{v_p}}} = {h \over {{m_\alpha }{v_\alpha }}}$$

$$ \therefore $$ $${{{v_p}} \over {{v_\alpha }}} = {{{m_\alpha }} \over {{m_p}}}$$ = $${{4{m_p}} \over {{m_p}}} = {4 \over 1}$$

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