Assertion (A): Two photons having equal wavelengths have equal linear momentum.
Reason (R): When monochromatic light shows its photon character, each photon has a linear momentum \(p = \frac{h}{\lambda}\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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The linear momentum of a photon is given by \(p = h/\lambda\). If two photons have equal wavelengths \(\lambda\), then their linear momenta \(p\) must also be equal. The Reason (R) correctly states the formula and explains Assertion (A).
Assertion (A): A photon and an electron both have energy \(50\text{ eV}\). Both have different wavelengths.
Reason (R): Wavelength depends on energy and not on mass.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For a photon, wavelength is \(\lambda_p = hc/E\). For an electron, de Broglie wavelength is \(\lambda_e = h/\sqrt{2mE}\). Since their formulas are different and \(\lambda_e\) depends on mass \(m\), their wavelengths will be different for the same energy. So (A) is true. Reason (R) is false because an electron's de Broglie wavelength depends on its mass.
Assertion (A): In photoelectric effect, cathode or emitter plate is usually coated with barium oxide, barium sulphide or strontium oxide.
Reason (R): Coating prevents cathode from erosion.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Cathodes are coated with materials like barium oxide to lower their work function, enhancing photoemission efficiency. So (A) is true. Coatings can indeed prevent erosion, so (R) is also true. However, preventing erosion is not the primary reason for choosing these specific low work function materials, so (R) is not the correct explanation for (A).
Assertion (A): A particle at rest breaks into two particles of different masses. They fly off in different directions. Their de Broglie wavelengths will be different.
Reason (R): Their speed will be different.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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When a particle at rest decays, conservation of momentum dictates that the two resulting particles must have equal and opposite momenta \( |p_1| = |p_2| \). Since the de Broglie wavelength is \( \lambda = h/p \), both particles must have the same wavelength, making the assertion that they are different false.
Assertion (A): In photon-particle collision the total energy and total momentum are conserved.
Reason (R): The number of photons are conserved in a collision.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Conservation laws in collisions.
Formula: Total energy \(E_{\text{total}}\) and total momentum \(p_{\text{total}}\) are conserved in all collisions.
Solution: In photon-particle collisions, total energy and momentum are conserved. However, photons can be absorbed or emitted, so their number is not necessarily conserved. Hence, Assertion (A) is true, but Reason (R) is false.
Assertion (A): The stopping potential increases, when frequency of incident rays are increased.
Reason (R): Stopping potential is directly proportional to the frequency of incident radiation.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Einstein's photoelectric equation.
Formula: \(eV_s = hf - \phi_0\) or \(V_s = \frac{h}{e}f - \frac{\phi_0}{e}\) where \(f\) is frequency and \(\phi_0\) is work function.
Solution: From the formula, as frequency \(f\) increases, the stopping potential \(V_s\) also increases. So (A) is true. However, \(V_s\) is linearly dependent on \(f\) with an intercept of \(-\frac{\phi_0}{e}\) (unless \(\phi_0 = 0\)), not directly proportional. So (R) is false.
Assertion (A): A metallic surface is irradiated by monochromatic light of frequency \(nu > nu_0\) (the threshold frequency). The maximum kinetic energy and stopping potential are \(K_{\text{max}}\) and \(V_s\) respectively. If the frequency of incident on the surface is doubled, both \(K_{\text{max}}\) and \(V_s\) are more than doubled.
Reason (R): The maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of incident light.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Photoelectric effect and linear dependence.
Formula: \(K_{\text{max}} = h\nu - h\nu_0\) and \(V_s = \frac{h\nu}{e} - \frac{h\nu_0}{e}\) where \(h\nu_0\) is the work function.
Solution: Both \(K_{\text{max}}\) and \(V_s\) are linearly dependent on frequency \(\nu\) with a positive slope and a negative intercept (work function term). Due to this negative intercept, if \(\nu\) is doubled, \(K_{\text{max}}\) and \(V_s\) will increase by more than double. Thus, both (A) and (R) are true, and (R) correctly explains (A).
Assertion (A): When ultraviolet light incident on a photo cell, its stopping potential is \(V_S\) and the maximum kinetic energy of photoelectrons is \(K_{\text{max}}\) . When the ultraviolet light is replaced by X-rays, both \(V_S\) and \(K_{\text{max}}\) increases.
Reason (R): Photo electrons are emitted with speed ranging from zero to a maximum value because of the range of frequencies present in the incident light.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Photoelectric effect and photon energy.
Formula: \(K_{\text{max}} = hf - \phi\) and \(eV_s = K_{\text{max}}\) where \(f\) is frequency.
Solution: X-rays have higher frequency and thus higher photon energy than ultraviolet light. Therefore, incident X-rays will produce photoelectrons with higher maximum kinetic energy (\(K_{\text{max}}\) ) and higher stopping potential (\(V_s\) ). So (A) is true. The range of photoelectron speeds is primarily due to energy losses as electrons travel through the material, not necessarily due to a range of frequencies in the incident light. So (R) is false.
Assertion (A): By de-Broglie hypothesis, \(p = h/\lambda\) for both the electron and the photon.
Reason (R): If an electron has the same wavelength as a photon, they have the same energy.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: De Broglie wavelength and energy relations.
Formula: De Broglie wavelength \(\lambda = h/p\). Photon energy \(E_p = pc = hc/\lambda\). Electron kinetic energy \(E_e = p^2/(2m) = h^2/(2m\lambda^2)\) (non-relativistic).
Solution: De Broglie's hypothesis states that momentum \(p = h/\lambda\) applies to all particles, including electrons and photons. So (A) is true. If an electron and a photon have the same wavelength, their energies are \(E_p = hc/\lambda\) and \(E_e = h^2/(2m\lambda^2)\), which are generally not equal. So (R) is false.
Assertion (A): Charge of a photon is zero.
Reason (R): Rest mass of a photon is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Concept: Fundamental properties of photons.
Formula: Energy-momentum relation \(E^2 = p^2c^2 + m_0^2c^4\). For a photon, \(m_0 = 0\).
Solution: Photons are the quanta of electromagnetic radiation and do not carry any electric charge. So (A) is true. Photons are massless particles, meaning their rest mass is zero. So (R) is true. The zero rest mass of a photon is intrinsically linked to its inability to carry charge and its nature as a mediator of the electromagnetic force. A charged particle must possess non-zero invariant mass for a consistent description in physics. Therefore, (R) provides a fundamental explanation for (A).