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): Cut-off wavelength of x-ray is independent of type of target metal
Reason (R): Wavelength of \(K_{alpha}\) x-ray depends upon type of target metal.
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: X-ray production mechanisms.
Formula: Cut-off wavelength \(\lambda_{\text{min}} = \frac{hc}{eV}\) (depends on voltage). Characteristic X-ray energy \(E = h\nu\) (depends on atomic transitions).
Solution: The cut-off (minimum) wavelength of continuous X-rays depends only on the accelerating voltage, not the target material. So (A) is true. \(K_{\alpha}\) X-rays are characteristic X-rays, whose wavelengths are specific to the target material. So (R) is true. However, (R) explains characteristic X-rays, not the cut-off wavelength; thus, (R) is not a correct explanation for (A).
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).
Assertion (A): The relative velocity of two photons travelling in opposite directions is \(c\).
Reason (R): The 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: Special Relativity, velocity addition formula.
Formula: Relativistic velocity addition \(v_{\text{rel}} = \frac{v_1 - v_2}{1 - \frac{v_1 v_2}{c^2}}\) where \(v_1, v_2\) are velocities of two objects.
Solution: For two photons moving in opposite directions (\(v_1 = c, v_2 = -c\)), the relativistic velocity addition formula gives \(v_{\text{rel}} = \frac{c - (-c)}{1 - \frac{c(-c)}{c^2}} = \frac{2c}{1+1} = c\). So (A) is true. The rest mass of a photon is zero, which is the reason why photons always travel at the speed of light \(c\) in all inertial frames, forming the foundation of special relativity. Thus, (R) explains the relativistic behavior described in (A).
Assertion (A): Work function of a metal increases with increase in intensity of incident light.
Reason (R): Maximum kinetic energy of ejected photoelectrons depends upon the intensity of incident light.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A) is false: Work function is a characteristic property of the metal and is independent of light intensity.
Reason (R) is false: Maximum kinetic energy depends on frequency, not intensity. Both statements are false.
Assertion (A): In the process of photo electric emission, all the emitted photoelectrons have same KE.
Reason (R): According to Einstein’s photo electric equation \(KE= h\nu – \phi\).
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A) is false: Kinetic energy of emitted photoelectrons varies from zero to maximum. Reason (R) is false: Einstein's equation is \(KE_{max}= h\nu - \phi\). Both statements are false.