The energies of two photons are in the ratio 1:4. The corresponding ratio of their momenta is a) 1:2 b) 1:4 c) 2:1 d) 4:1

To solve the problem regarding the ratio of momenta of two photons given their energy ratio of 1:4, we need to understand the relationship between energy and momentum in the context of photons. Photons, which are particles of light, have a direct correlation between their energy (E) and momentum (p), described by the equations from quantum physics.

Fundamental Relationships

The energy of a photon is given by the equation:

E = hf

where h is Planck's constant and f is the frequency of the photon. The momentum of a photon can be expressed as:

p = \frac{E}{c}

where c is the speed of light. Therefore, momentum is directly proportional to energy for photons. This means we can represent momentum as:

p = \frac{hf}{c}

Analyzing the Problem

Given that the energies of two photons are in the ratio 1:4, we can represent their energies as:

• Photon 1: E1 = E
• Photon 2: E2 = 4E

Now, using the relationship between energy and momentum, we can calculate their momenta:

• For Photon 1: p1 = E/c = E/c
• For Photon 2: p2 = E2/c = 4E/c

Calculating the Ratio of Momenta

Now, we can find the ratio of their momenta:

\frac{p1}{p2} = \frac{E/c}{4E/c} = \frac{1}{4}

This tells us that the ratio of momenta of the two photons is 1:4.

Final Answer

From the options provided, the correct answer corresponds to the momenta ratio of 1:4, which means the answer is d) 4:1.

In summary, when dealing with photons, always remember the direct proportionality between energy and momentum, which simplifies solving these types of problems significantly. If you have any further questions or need clarification on any part, feel free to ask!