Choose correct option :
(A) Number of photons required for a light beam of 2000 pm wavelength and 1 Joule energy is \(1.01 \times 10^{16}\).
(B) Light with wavelength 300 nm has energy \(E_1\) and for wavelength 900 nm has energy \(E_2\), then \(E_1/E_2 = 1/3\).
(C) Frequency of light is \(4.5 \times 10^{16}\) Hz then its wavelength is \(6.7 \times 10^{-9}\) m.
(D) If electrons and protons are accelerated by same potential difference, then their de-Broglie wavelength are equal.
(A) Number of photons required for a light beam of 2000 pm wavelength and 1 Joule energy is \(1.01 \times 10^{16}\).
(B) Light with wavelength 300 nm has energy \(E_1\) and for wavelength 900 nm has energy \(E_2\), then \(E_1/E_2 = 1/3\).
(C) Frequency of light is \(4.5 \times 10^{16}\) Hz then its wavelength is \(6.7 \times 10^{-9}\) m.
(D) If electrons and protons are accelerated by same potential difference, then their de-Broglie wavelength are equal.
Show Hint
Remember the relationship \(\lambda = h/p\) and how momentum \(p\) relates to kinetic energy \(K\) (\(p=\sqrt{2mK}\)) and accelerating potential \(V\) (\(K=qV\)). This is a very common topic.
- A only
- A & B only
- A & C only
- A, B, C & D
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We need to evaluate the correctness of four separate statements related to photons, electromagnetic waves, and de-Broglie wavelength.
Step 2: Detailed Explanation:
Statement (A):
Energy of a single photon is \(E_{ph} = \frac{hc}{\lambda}\).
The number of photons \(N\) in a beam of total energy \(E_{total}\) is \(N = \frac{E_{total}}{E_{ph}} = \frac{E_{total} \lambda}{hc}\).
Given: \(E_{total} = 1\) J, \(\lambda = 2000 \text{ pm} = 2000 \times 10^{-12}\) m.
\(h = 6.626 \times 10^{-34}\) J·s, \(c = 3 \times 10^8\) m/s.
\[ N = \frac{(1 \text{ J})(2000 \times 10^{-12} \text{ m})}{(6.626 \times 10^{-34} \text{ Js})(3 \times 10^8 \text{ m/s})} = \frac{2 \times 10^{-9}}{19.878 \times 10^{-26}} \approx 0.1006 \times 10^{17} = 1.006 \times 10^{16} \] This value is very close to \(1.01 \times 10^{16}\). Statement (A) is correct.
Statement (B):
The energy of a photon is inversely proportional to its wavelength (\(E \propto 1/\lambda\)).
Therefore, the ratio of energies is \(\frac{E_1}{E_2} = \frac{\lambda_2}{\lambda_1}\).
Given: \(\lambda_1 = 300\) nm, \(\lambda_2 = 900\) nm.
\[ \frac{E_1}{E_2} = \frac{300 \text{ nm}}{900 \text{ nm}} = 3 \] The statement claims \(\frac{E_1}{E_2} = 1/3\). Statement (B) is incorrect.
Statement (C):
The relationship between frequency (\(f\)), wavelength (\(\lambda\)), and the speed of light (\(c\)) is \(\lambda = \frac{c}{f}\).
Given: \(f = 4.5 \times 10^{16}\) Hz.
\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{4.5 \times 10^{16} \text{ Hz}} = \frac{3}{4.5} \times 10^{-8} \text{ m} = \frac{2}{3} \times 10^{-8} \text{ m} \approx 0.667 \times 10^{-8} \text{ m} = 6.67 \times 10^{-9} \text{ m} \] This value is approximately \(6.7 \times 10^{-9}\) m. Statement (C) is correct.
Statement (D):
The de-Broglie wavelength is given by \(\lambda = \frac{h}{p}\), where \(p\) is the momentum.
When a particle of charge \(q\) is accelerated by a potential difference \(V\), its kinetic energy is \(K = qV\).
Momentum is related to kinetic energy by \(p = \sqrt{2mK}\).
So, \(\lambda = \frac{h}{\sqrt{2mK}} = \frac{h}{\sqrt{2mqV}}\).
Electrons and protons have the same magnitude of charge \(q\) but different masses (\(m_p >> m_e\)). If they are accelerated by the same \(V\), their wavelengths will be different. Specifically, since mass is in the denominator, the more massive proton will have a shorter wavelength. Statement (D) is incorrect.
Step 3: Final Answer:
Statements (A) and (C) are correct, while (B) and (D) is incorrect. The best choice among the options is (C) which states A & C are correct.
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Dual nature of matter Questions
- A convex lens of focal length 40 cm forms an image of an extended source of light on a photo-electric cell. A current \( I \) is produced. The lens is replaced by another convex lens having the same diameter but focal length 20 cm. The photoelectric current now is:
- When a metal surface is illuminated by light of wavelength \( \lambda \), the stopping potential is 8V. When the same surface is illuminated by light of wavelength \( 3\lambda \), the stopping potential is 2V. The threshold wavelength for this surface is:
- A new unit (\(\alpha\)) of length is chosen such that it is equal to the distance travelled by light in vacuum in 1 second. What is the distance between Venus and Earth in terms of this new unit. If light takes 6 min 40 sec to cover the distance.
- Two photons of wavelength \( \lambda \) and \(2\lambda\) are incident on a metal surface and emit photoelectrons of maximum kinetic energies \(3k\) and \(k\). Find work function of the metal.
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.