The ratio of wavelength of a photon and that of an electron of same energy E will be

Options

A. `sqrt((2m)/(E))`
B. `sqrt((E)/(2m))`
C. `sqrt((2m)/(E))c`
D. `sqrt((EC)/(2m))`

Answer

To find the ratio of the wavelength of a photon to that of an electron with the same energy \( E \), we can follow these steps:

### Step 1: Determine the Wavelength of a Photon
The energy of a photon is given by the equation:
\[ E = h \nu \]
where \( h \) is Planck's constant and \( \nu \) is the frequency of the photon. The relationship between frequency and wavelength is given by:
\[ \nu = \frac{c}{\lambda} \]
where \( c \) is the speed of light and \( \lambda \) is the wavelength. Substituting this into the energy equation gives:
\[ E = \frac{hc}{\lambda} \]
Rearranging this for the wavelength of the photon, we get:
\[ \lambda_p = \frac{hc}{E} \]

### Step 2: Determine the Wavelength of an Electron
For an electron, we can use the de Broglie wavelength formula:
\[ \lambda_e = \frac{h}{p} \]
where \( p \) is the momentum of the electron. The momentum of an electron can be expressed in terms of its kinetic energy \( E \) as:
\[ p = \sqrt{2mE} \]
where \( m \) is the mass of the electron. Substituting this into the de Broglie wavelength formula gives:
\[ \lambda_e = \frac{h}{\sqrt{2mE}} \]

### Step 3: Calculate the Ratio of Wavelengths
Now we can find the ratio of the wavelengths of the photon and the electron:
\[ \frac{\lambda_p}{\lambda_e} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}} \]
This simplifies to:
\[ \frac{\lambda_p}{\lambda_e} = \frac{hc}{E} \cdot \frac{\sqrt{2mE}}{h} \]
The \( h \) cancels out:
\[ \frac{\lambda_p}{\lambda_e} = \frac{c \sqrt{2mE}}{E} \]
Thus, we can express this as:
\[ \frac{\lambda_p}{\lambda_e} = c \cdot \frac{\sqrt{2m}}{\sqrt{E}} \]

### Final Result
The ratio of the wavelength of a photon to that of an electron with the same energy \( E \) is:
\[ \frac{\lambda_p}{\lambda_e} = c \sqrt{\frac{2m}{E}} \]