An electron is confined to a tube of length L. The electron’s potential energy in one half of the tube is zero, while the potential energy in the other half is 10eV. If the electron has a total energy `E = 15 eV`, then the ratio of the deBroglie wavelength of the electron in the 10eV region of the tube to that in the other half is -

Options

A. `1//sqrt3`
B. `sqrt3`
C. 3
D. `(1)/(3)`

Answer

To solve the problem, we need to find the ratio of the de Broglie wavelengths of an electron in two different potential energy regions of a tube. The potential energy in one half of the tube is 0 eV, while in the other half it is 10 eV. The total energy of the electron is given as 15 eV.

### Step-by-step Solution:

1. **Identify the Kinetic Energy in Each Region:**
- In the region where the potential energy (U) is 0 eV:
- Total Energy (E) = Kinetic Energy (K) + Potential Energy (U)
- Here, U = 0 eV, so:
\[ E = K + 0 \Rightarrow K = E = 15 \text{ eV} \]

- In the region where the potential energy (U) is 10 eV:
- Again using the total energy equation:
\[ E = K + U \Rightarrow 15 \text{ eV} = K + 10 \text{ eV} \]
- Solving for K:
\[ K = 15 \text{ eV} - 10 \text{ eV} = 5 \text{ eV} \]

2. **Calculate the de Broglie Wavelength:**
- The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{p} \]
- The momentum (p) can be expressed in terms of kinetic energy (K):
\[ p = \sqrt{2mK} \]
- Thus, the de Broglie wavelength becomes:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]

3. **Find the Wavelengths in Each Region:**
- For the region with U = 0 eV (K = 15 eV):
\[ \lambda_0 = \frac{h}{\sqrt{2m \cdot 15 \text{ eV}}} \]

- For the region with U = 10 eV (K = 5 eV):
\[ \lambda_{10} = \frac{h}{\sqrt{2m \cdot 5 \text{ eV}}} \]

4. **Calculate the Ratio of the Wavelengths:**
- The ratio of the de Broglie wavelengths is:
\[ \frac{\lambda_{10}}{\lambda_0} = \frac{\frac{h}{\sqrt{2m \cdot 5}}}{\frac{h}{\sqrt{2m \cdot 15}}} = \frac{\sqrt{15}}{\sqrt{5}} = \sqrt{\frac{15}{5}} = \sqrt{3} \]

### Final Answer:
The ratio of the de Broglie wavelength of the electron in the 10 eV region to that in the 0 eV region is \(\sqrt{3}\).