Two particles A and B of same mass have their de - Broglie wavelength in the ratio `X_(A):X_(B)=K:1`. Their potential energies `U_(A):U_(B)=1:K^(2)`. The ratio of their total energies `E_(A):E_(B)` is

Options

A. `K^(2):1`
B. `1:K^(2)`
C. `K:1`
D. `1:K`

Answer

To solve the problem, we need to find the ratio of the total energies of two particles A and B given their de Broglie wavelengths and potential energies.

### Step-by-Step Solution:

1. **Understanding the Given Ratios**:
- The de Broglie wavelengths of particles A and B are in the ratio \( \lambda_A : \lambda_B = K : 1 \).
- The potential energies are in the ratio \( U_A : U_B = 1 : K^2 \).

2. **Expressing Potential Energies**:
- Let \( U_A = y \) (some arbitrary value).
- Then, from the ratio, we have \( U_B = K^2 y \).

3. **Using the de Broglie Wavelength Formula**:
- The de Broglie wavelength is given by:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]
- From the ratio of wavelengths, we can express the kinetic energies \( K_A \) and \( K_B \) in terms of their wavelengths.

4. **Relating Kinetic Energies**:
- The kinetic energy \( K \) is related to the de Broglie wavelength:
\[ K = \frac{h^2}{2m\lambda^2} \]
- Therefore, for particles A and B:
\[ K_A = \frac{h^2}{2m\lambda_A^2} \quad \text{and} \quad K_B = \frac{h^2}{2m\lambda_B^2} \]
- Substituting the ratios:
\[ K_A = \frac{h^2}{2m(K^2)} \quad \text{and} \quad K_B = \frac{h^2}{2m(1)} \]
- This gives us:
\[ K_A : K_B = \frac{1}{K^2} : 1 \]

5. **Expressing Total Energies**:
- The total energy \( E \) of a particle is the sum of its kinetic and potential energies:
\[ E_A = K_A + U_A = \frac{1}{K^2} + y \]
\[ E_B = K_B + U_B = 1 + K^2y \]

6. **Finding the Ratio of Total Energies**:
- Now, we need to find the ratio \( E_A : E_B \):
\[ E_A : E_B = \left(\frac{1}{K^2} + y\right) : \left(1 + K^2y\right) \]
- To simplify, we can express this as:
\[ E_A : E_B = \frac{1 + K^2y}{K^2(1 + K^2y)} \]
- This simplifies to:
\[ E_A : E_B = \frac{1}{K^2} \]
- Therefore, the final ratio is:
\[ E_A : E_B = 1 : K^2 \]

### Final Answer:
The ratio of their total energies \( E_A : E_B \) is \( 1 : K^2 \).