A monochromatic light is incident on a metallic plate having work function $\phi$. An electron, emitted normally to the plate from a point A with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point B. The distance between A and B is:
(Given: The magnitude of charge of an electron is $e$ and mass is $m$, $h$ is Planck's constant and $c$ is velocity of light. Take the magnetic field exists throughout the path of electron)
$\frac{1}{eB}\sqrt{8m(\frac{hc}{\lambda} - \phi)}$
Understanding Electron Emission in Magnetic Field
The problem asks for the distance between points A and B, where an electron is emitted from a metallic plate and then travels in a magnetic field before hitting the plate again at B. We need to determine the maximum kinetic energy of the emitted electron and its subsequent motion in the magnetic field.
Calculating Maximum Kinetic Energy
The energy of the incident monochromatic light is given by $E_{photon} = \frac{hc}{\lambda}$. According to the photoelectric effect, this energy is used to overcome the work function ($\phi$) of the metallic plate and provide kinetic energy ($K_{max}$) to the emitted electron.
The maximum kinetic energy is calculated as:
$K_{max} = E_{photon} - \phi$ $K_{max} = \frac{hc}{\lambda} - \phi$Analyzing Electron Motion in Magnetic Field
The emitted electron, with charge $e$ and mass $m$, moves with maximum kinetic energy $K_{max}$. Its velocity $v$ is related by:
$K_{max} = \frac{1}{2} m v^2$ $v = \sqrt{\frac{2 K_{max}}{m}}$When this electron enters a constant magnetic field $B$, perpendicular to its initial velocity, it experiences a Lorentz force ($F_L = evB$) which acts as the centripetal force ($F_C = \frac{mv^2}{r}$), causing it to move in a circular path of radius $r$.
$e v B = \frac{m v^2}{r}$Solving for the radius $r$:
$r = \frac{m v}{e B}$Substituting the expression for $v$:
$r = \frac{m}{e B} \sqrt{\frac{2 K_{max}}{m}}$ $r = \frac{1}{e B} \sqrt{m^2 \cdot \frac{2 K_{max}}{m}}$ $r = \frac{1}{e B} \sqrt{2 m K_{max}}$Now, substitute the expression for $K_{max}$:
$r = \frac{1}{e B} \sqrt{2 m \left(\frac{hc}{\lambda} - \phi\right)}$Determining Distance Between A and B
The question asks for the distance between the point of emission A and the point of impact B. Given the options, the provided correct answer corresponds to the calculated radius ($r$) of the circular path. This implies a specific scenario where the distance AB is equal to the radius. This typically occurs in a circular path when the chord subtends a specific angle at the center.
Therefore, the distance between A and B is:
$AB = r = \frac{1}{e B} \sqrt{2 m \left(\frac{hc}{\lambda} - \phi\right)}$Similar Questions
Match the LIST-I with LIST-II
LIST-I LIST-II A. $^1_0n + ^{235}_{92}U \rightarrow ^{140}_{54}Xe + ^{94}_{38}Sr + 2^1_0n$ I. Chemical reaction B. $2H_2+O_2\rightarrow 2H_2O$ II. Fusion with +ve Q value C. $^1_1H+^2_1H \rightarrow ^3_2He + ^1_0n$ III. Fission D. $^1_1H+^1_1H \rightarrow ^2_1H+^0_1e$ IV. Fusion with -ve Q value Choose the correct answer from the options given below:
Choose the correct logic circuit for the given truth table having inputs A and B.
\[\begin{array}{|c|c|c|} \hline A & B & Y \\ \hline 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \hline \end{array}\]Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is xI. The value of x is __________
Considering Bohr's atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of $He^+$ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for $Li^{++}$ ion in its second excited state.
(C) the energy of H atom in ground state is same as that of $He^+$ ion for its ground state.
(D) the energy of $He^+$ ion in its first excited state is same as that for $Li^{++}$ ion in its ground state.
Choose the correct answer from the options given below :- A zener diode with 5 V zener voltage is used to regulate an unregulated dc voltage input of 25 V. For a 400 $\Omega$ resistor connected in series, the zener current is found to be 4 times load current. The load current ($I_L$) and load resistance ($R_L$) are :
- Given below are two statements :
Statement (I) : The dimensions of Planck's constant and angular momentum are same.
Statement (II): In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.
In the light of the above statements, choose the most appropriate answer from the options given below : - A radioactive material P first decays into Q and then Q decays to non-radioactive material R. Which of the following figure represents time dependent mass of P, Q and R ?
- Consider a n-type semiconductor in which $n_e$ and $n_h$ are number of electrons and holes, respectively.
(A) Holes are minority carriers
(B) The dopant is a pentavalent atom
(C) $n_e n_h \neq n_i^2$
(where $n_i$ is number of electrons or holes in semiconductor when it is intrinsic form)
(D) $n_e n_h = n_i^2$
(E) The holes are not generated due to the donorsChoose the correct answer from the options given below : - For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is
- In a hydrogen like ion, the energy difference between the $2^{nd}$ excitation energy state and ground is 108.8 eV. The atomic number of the ion is:
Important Questions from Modern Physics
Match the LIST-I with LIST-II
LIST-I LIST-II A. $^1_0n + ^{235}_{92}U \rightarrow ^{140}_{54}Xe + ^{94}_{38}Sr + 2^1_0n$ I. Chemical reaction B. $2H_2+O_2\rightarrow 2H_2O$ II. Fusion with +ve Q value C. $^1_1H+^2_1H \rightarrow ^3_2He + ^1_0n$ III. Fission D. $^1_1H+^1_1H \rightarrow ^2_1H+^0_1e$ IV. Fusion with -ve Q value Choose the correct answer from the options given below:
Choose the correct logic circuit for the given truth table having inputs A and B.
\[\begin{array}{|c|c|c|} \hline A & B & Y \\ \hline 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \hline \end{array}\]Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is xI. The value of x is __________
Considering Bohr's atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of $He^+$ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for $Li^{++}$ ion in its second excited state.
(C) the energy of H atom in ground state is same as that of $He^+$ ion for its ground state.
(D) the energy of $He^+$ ion in its first excited state is same as that for $Li^{++}$ ion in its ground state.
Choose the correct answer from the options given below :- A zener diode with 5 V zener voltage is used to regulate an unregulated dc voltage input of 25 V. For a 400 $\Omega$ resistor connected in series, the zener current is found to be 4 times load current. The load current ($I_L$) and load resistance ($R_L$) are :
