Photoelectric Effect

From Einstein’s equation: Using both cases and converting units, after simplifying, we find:

Energy of one photon: Number of photons per second:

For electrons to be ejected from a metal surface (photoelectric effect), the energy of the incident photons ( ) must be greater than or equal to the work function ( ) of the metal. . If , electrons are not ejected. First, calculate the energy of the incident photons. The energy of a photon is given by , where is Planck's constant, is the speed of light, and is the wavelength of the radiation. Given values: Wavelength . Planck's constant . (The image shows Js, typically Js is used). Speed of light . Calculate in Joules: . . . . So, . Convert to electron-volts (eV), since work functions are given in eV. . (Using J for simplicity if not specified). . (Using for better precision). . A common shortcut for photon energy in eV when wavelength is in nm is or . Using : . Or using . . So 1240 is good. . Now compare with the work functions ( ) of the metals: \begin{itemize} \item Li: . Since , electrons are ejected. \item Mg: . Since , electrons are NOT ejected. \item Cu: . Since , electrons are NOT ejected. \item Ag: . Since , electrons are NOT ejected. \item K: . Since , electrons are ejected. \item Na: . Since , electrons are ejected. \end{itemize} The metals that do NOT eject electrons are Mg, Cu, and Ag. There are 3 such metals. This matches option (c).

To "just eject" electrons, the energy of the incident photons ( ) must be equal to the work function ( ) of the metal. This corresponds to the threshold condition. . The energy of a photon is given by , where is Planck's constant, is the speed of light, and is the wavelength. So, . This gives the threshold wavelength . Given values: Work function . We need to find in nm. A useful formula for this conversion is . The value of is approximately or . Let's use . . . Calculation: . : So, . Let's try with : . So, . Comparing with the options: % Option (a) 102 nm % Option (b) 330 nm % Option (c) 197 nm % Option (d) 310 nm The calculated value is closest to option (c) 197 nm. This suggests or a similar value was used. (If using Js, m/s, C, then eV nm). nm. If Js: eV nm. nm. Option (c) 197 nm is the best fit.

Work function is the minimum energy required to eject an electron from the metal surface. Platinum has a very high work function (~5.6 eV), whereas Cesium and Sodium have low values (~2.1 eV and ~2.3 eV respectively), and Aluminium is moderate (~4.3 eV).

Step 1: Use the photoelectric effect equation.
Step 2: Find the frequencies for H and H lines (Balmer series). For H ( ): For H ( ):
Step 3: Substitute into the kinetic energy formula. Find a common denominator: Final Answer:

The de Broglie wavelength of a particle is related to its momentum by: Rearranging for : Substitute the given values:

The units are valid: , which is correct for momentum. Hence, the correct answer is option (3).

According to the photoelectric equation: where is the kinetic energy of the most energetic photoelectron, is Planck's constant, and is the work function of the metal. From de Broglie's equation for the wavelength of a particle, we have: where is the momentum of the electron, and momentum . Substitute for from the photoelectric equation: Now, replacing with , you will get the final expression for the de Broglie wavelength in terms of and , matching option (2). Thus, the correct answer is option (2), .

The work function of the material is related to the change in kinetic energy of the emitted photoelectrons when the wavelength is halved, as described by Einstein's photoelectric equation. Thus, the correct answer is option (2).

According to Einstein’s photoelectric equation: For frequency : Now, for frequency : Substitute : Thus, the new velocity is , matching option (3).

According to the photoelectric equation: Given:
- Energy of each photon =
- Work function
Even if two photons arrive simultaneously, photoelectric effect requires single photon interaction per electron. So only is available, which is less than the work function.
Hence, no photoelectrons are emitted, and:

According to Einstein's photoelectric equation, the maximum kinetic energy of the emitted photoelectrons is given by: $ is Planck's constant, is the frequency of the incident light, and is the work function of the material.
In the first case, the frequency of the incident light is , and the maximum kinetic energy is .
So, we have: \) .
Let the maximum kinetic energy of the emitted photoelectrons in this case be .
Then, according to Einstein's photoelectric equation: \) in terms of and .
From equation (1), we can express as: \) into equation (2): \) $ Thus, if the frequency of the incident light is , the maximum kinetic energy of the emitted photoelectrons is .

According to the photoelectric equation: where is the kinetic energy of the emitted photoelectrons, is the frequency of the incident photons, and is the threshold frequency. The maximum linear momentum is related to the kinetic energy by the equation: where is the mass of the photoelectron. For photoelectrons to have maximum linear momentum , the frequency of the incident photons must be increased. By analyzing the relationship between frequency and momentum, we determine that the frequency of the photons should be to achieve the required momentum. Thus, the correct answer is .

Step 1: Understanding the photoelectric effect.
The photo current in a photo diode is primarily influenced by the frequency of the incident light. According to the photoelectric effect, when light of a certain frequency strikes a metal surface, electrons are emitted, creating a photo current. The intensity of the light affects the number of emitted electrons, while the frequency determines if the electrons will have enough energy to be emitted at all.
Step 2: Answer Justification.
The frequency of the incident light is the critical factor that determines the emission of electrons. If the frequency is below a certain threshold, no current is generated, regardless of the light's intensity. This is why the correct answer is option (2).

Step 1: Apply Energy and Photoelectric Equation The kinetic energy of the emitted photoelectron is given by: Since , the velocity is given by: Step 2: Compute Kinetic Energy For : For : Step 3: Compute Velocity Ratio Thus, the correct answer is .

Step 1: de Broglie Wavelength Formula The de Broglie wavelength for a particle accelerated through a potential difference is: where: - is Planck’s constant, - is the mass of the particle, - is the kinetic energy gained due to acceleration. Step 2: Applying the Formula to Proton and Alpha Particle For a proton: For an alpha particle: - Mass , - Charge . Since they are accelerated through the same potential , the kinetic energy of the alpha particle is: Thus, the ratio of de Broglie wavelengths: Conclusion Thus, the correct answer is:

Perfect elastic collision, one dimensional
Using conservation of momentum and energy
Final velocity of particle:
Final velocity of particle:
de Broglie wavelength
So

Step 1: de Broglie Wavelength Formula The de Broglie wavelength for a particle accelerated through a potential difference is: where: - is Planck’s constant, - is the mass of the particle, - is the kinetic energy gained due to acceleration. Step 2: Applying the Formula to Proton and Alpha Particle For a proton: For an alpha particle: - Mass , - Charge . Since they are accelerated through the same potential , the kinetic energy of the alpha particle is: Thus, the ratio of de Broglie wavelengths: Conclusion Thus, the correct answer is:

Step 1: de Broglie wavelength relation
where is momentum.
Step 2: Calculate change in wavelength
Given: Since , small change in due to change in is: But as momentum changes by , . So: Step 3: Conclusion
Initial momentum is .

Step 1: Use the formula for shortest wavelength in any series: where is the principal quantum number of the lower energy level of the series. Step 2: For Balmer series, Step 3: For Brackett series, Step 4: Take ratio of wavelengths (remember it's inverse of above)

The de Broglie wavelength is given by: $ h m k T$ is the temperature in Kelvin. The ratio of wavelengths depends on the square root of the inverse of the temperature ratio. Thus, the ratio of wavelengths at 127°C and 352°C is 5:3.

Let’s break this down step by step to calculate the maximum kinetic energy of the photoelectrons and determine why option (2) is the correct answer.
Step 1: Understand the photoelectric effect
The maximum kinetic energy of photoelectrons is given by Einstein’s photoelectric equation:

where:

• is the energy of the incident photon,
• is the work function of the material.

Step 2: Convert the photon energy to eV
Photon energy, .
1 eV = , so:

Work function, . This would mean no photoelectrons are emitted since . Assuming a typo, let’s test :

Step 3: Confirm the correct answer
With , the maximum kinetic energy is 2.5 eV, matching option (2).
Thus, the correct answer is (2) 2.5 eV.