An electron & a photon have same energy E. Find the ratio of de Broglie wavelength of electron to wavelength of photon. Given mass of electron is m & speed of light is C

Options

A. `c(2mE)^(1//2)`
B. `((E)/(2m))^(1//2)`
C. `(1)/(c)((E)/(2m))^(1//2)`
D. `(1)/(c)((2E)/(m))^(1//2)`

Answer

To solve the problem of finding the ratio of the de Broglie wavelength of an electron to the wavelength of a photon when both have the same energy \( E \), we can follow these steps:

### Step 1: Determine the wavelength of the photon
The energy of a photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength of the photon.

Rearranging this equation to find the wavelength of the photon, we have:
\[ \lambda_p = \frac{hc}{E} \]

### Step 2: Determine the de Broglie wavelength of the electron
The de Broglie wavelength \( \lambda_e \) of an electron is given by:
\[ \lambda_e = \frac{h}{p} \]
where \( p \) is the momentum of the electron.

The momentum \( p \) of the electron can be expressed in terms of its kinetic energy \( E \):
\[ E = \frac{p^2}{2m} \]
From this, we can express momentum \( p \) as:
\[ p = \sqrt{2mE} \]

Now substituting this expression for \( p \) into the de Broglie wavelength formula, we get:
\[ \lambda_e = \frac{h}{\sqrt{2mE}} \]

### Step 3: Find the ratio of the de Broglie wavelength of the electron to the wavelength of the photon
Now we need to find the ratio \( \frac{\lambda_e}{\lambda_p} \):
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{\sqrt{2mE}}}{\frac{hc}{E}} \]

This simplifies to:
\[ \frac{\lambda_e}{\lambda_p} = \frac{h}{\sqrt{2mE}} \cdot \frac{E}{hc} = \frac{E}{hc\sqrt{2mE}} = \frac{\sqrt{E}}{hc\sqrt{2m}} \]

### Final Result
Thus, the ratio of the de Broglie wavelength of the electron to the wavelength of the photon is:
\[ \frac{\lambda_e}{\lambda_p} = \frac{\sqrt{E}}{hc\sqrt{2m}} \]