Q: The allowed energy for the particle for a particular value of n is proportional to :

\$ a = \cfrac{n \lambda}{2}\$De broglie wavelength\$ \lambda = \cfrac{h}{mv}\$Subsitute\$ p =\cfrac{nh}{2a}\$\$ E = \cfrac{1}{2}mv^2= \cfrac{p^2}{2m} =\cfrac{n^2h^2}{8a^2m}\$

Q: A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30\$mW\$and the speed of light is \$3\times 10^{8}m/s\$. The final momentum of the object is:

Sol\$. (B)\$\$t=100\times 10^{-9} sec,\ P=30\times 10^{-3}\$Watt, \$C=3\times 10^{8}m/s\$Incident energy \$= Pt\$By debroglie wavelength, \$\lambda=h/p\$and Energy of photon is \$E=hc/\lambda\$From above two equations, \$E=pc\$ Momentum \$p=\dfrac{Pt}{C}=\dfrac{30\times 10^{-3}\times 100\times 10^{-9}}{3\times 10^{8}}=1.0\times 10^{-17}\$ kg ms\$^{-1} \$So, the answer is option (B).

Q: Light of wavelength \$\displaystyle { \lambda }_{ ph }\$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is \$\displaystyle \phi \$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is \$\displaystyle { \lambda }_{ e }\$, which of the following statement(s) is(are) true?

\${ \lambda }_{ e }=\$ minimum be broglie wavelength.\$\dfrac { hc }{ { \lambda }_{ ph } } =\dfrac { hc }{ { \lambda }_{ e } } +\phi +eV\$\$\Rightarrow\$ \$hc(\dfrac { -1 }{ { \lambda }_{ ph }^{ 2 } } d{ \lambda }_{ ph })=hc(\dfrac { -1 }{ { \lambda }_{ e }^{ 2 } } d{ \lambda }_{ e })\$\$\Rightarrow\$ \$\dfrac { d{ \lambda }_{ ph } }{ d{ \lambda }_{ e } } =\dfrac { { \lambda }_{ ph }^{ 2 } }{ { \lambda }_{ e }^{ 2 } } \$This denies option A, B and C.\$V>>\dfrac { \phi }{ e } \$Energy of electron,\$E=\dfrac { { P }^{ 2 } }{ 2m } \$\$\lambda_e =\dfrac { h }{ p } =\dfrac { h }{ \sqrt { 2mE } } \$\$=\dfrac{h}{\sqrt{2meV}}\$This supports option D.

Question 1

The allowed energy for the particle for a particular value of n is proportional to :

Accepted Answer

$ a = \cfrac{n \lambda}{2}$De broglie wavelength$ \lambda = \cfrac{h}{mv}$Subsitute$ p =\cfrac{nh}{2a}$$ E = \cfrac{1}{2}mv^2= \cfrac{p^2}{2m} =\cfrac{n^2h^2}{8a^2m}$

Question 2

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 $m W$ and the speed of light is $3 \times 1 0^{8} m / s$ . The final momentum of the object is:

Accepted Answer

Sol$. (B)$$t=100\times 10^{-9} sec,\ P=30\times 10^{-3}$Watt, $C=3\times 10^{8}m/s$Incident energy $= Pt$By debroglie wavelength, $\lambda=h/p$and Energy of photon is $E=hc/\lambda$From above two equations, $E=pc$ Momentum $p=\dfrac{Pt}{C}=\dfrac{30\times 10^{-3}\times 100\times 10^{-9}}{3\times 10^{8}}=1.0\times 10^{-17}$ kg ms$^{-1} $So, the answer is option (B).

Question 3

The speed of the particle that can take discrete values is proportional to

Accepted Answer

As given in the questionLet the no. of loops in the wave be  $n$Let wavelength be $\lambda$ Debroglie wavelength$ \lambda = \cfrac{h}{mv}$ ------------(1)Also$ n \lambda = 2 \pi a$  --------------(2)Substituting above$ v = \cfrac{nh}{2am}$$v \propto n$So, the answer is option (D).

Question 4

Light of wavelength $\displaystyle { \lambda  }_{ ph }$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\displaystyle \phi $ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is $\displaystyle { \lambda  }_{ e }$, which of the following statement(s) is(are) true?

Accepted Answer

${ \lambda  }_{ e }=$ minimum be broglie wavelength.$\dfrac { hc }{ { \lambda  }_{ ph } } =\dfrac { hc }{ { \lambda  }_{ e } } +\phi +eV$$\Rightarrow$ $hc(\dfrac { -1 }{ { \lambda  }_{ ph }^{ 2 } } d{ \lambda  }_{ ph })=hc(\dfrac { -1 }{ { \lambda  }_{ e }^{ 2 } } d{ \lambda  }_{ e })$$\Rightarrow$ $\dfrac { d{ \lambda  }_{ ph } }{ d{ \lambda  }_{ e } } =\dfrac { { \lambda  }_{ ph }^{ 2 } }{ { \lambda  }_{ e }^{ 2 } } $This denies option A, B and C.$V>>\dfrac { \phi  }{ e } $Energy of electron,$E=\dfrac { { P }^{ 2 } }{ 2m } $$\lambda_e =\dfrac { h }{ p } =\dfrac { h }{ \sqrt { 2mE }  } $$=\dfrac{h}{\sqrt{2meV}}$This supports option D.

Dual Nature Of Radiation And Matter

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The allowed energy for the particle for a particular value of n is proportional to :

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 $m W$ and the speed of light is $3 \times 1 0^{8} m / s$ . The final momentum of the object is:

The speed of the particle that can take discrete values is proportional to

Dual Nature Of Radiation And Matter

Physics 1478 Views

The allowed energy for the particle for a particular value of n is proportional to :

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30mWand the speed of light is 3×108m/s. The final momentum of the object is:

The speed of the particle that can take discrete values is proportional to

Physics
1478 Views

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 $m W$ and the speed of light is $3 \times 1 0^{8} m / s$ . The final momentum of the object is: