If m is the mass of an electron and c the speed of light, the ratio of the wavelength of a photon of energy E to that of the electron of the same energy is

Options

A. `csqrt((2m)/(E))`
B. `sqrt((2m)/(E))`
C. `sqrt((2m)/(cE))`
D. `sqrt((m)/(E))`

Explanation

To find the ratio of the wavelength of a photon (λ_p) to that of an electron (λ_e) when both have the same energy (E), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy-Wavelength Relationship for Photons:**
The energy (E) of a photon is given by the equation:
\[ E = \frac{hc}{\lambda_p} \]
where \(h\) is Planck's constant and \(c\) is the speed of light.

2. **Rearrange the Equation for Photon Wavelength:**
From the above equation, we can express the wavelength of the photon (λ_p) as:
\[ \lambda_p = \frac{hc}{E} \]

3. **Understand the Energy-Wavelength Relationship for Electrons:**
The energy (E) of an electron can also be expressed using its wavelength. The de Broglie wavelength (λ_e) of an electron is given by:
\[ E = \frac{h}{\lambda_e} \cdot \frac{1}{2m} \]
Rearranging gives us:
\[ \lambda_e = \frac{h}{\sqrt{2mE}} \]
where \(m\) is the mass of the electron.

4. **Set Up the Ratio of Wavelengths:**
Now, we want to find the ratio of the wavelengths:
\[ \frac{\lambda_p}{\lambda_e} = \frac{\frac{hc}{E}}{\frac{h}{\sqrt{2mE}}} \]

5. **Simplify the Ratio:**
Simplifying the above expression:
\[ \frac{\lambda_p}{\lambda_e} = \frac{hc}{E} \cdot \frac{\sqrt{2mE}}{h} \]
The \(h\) cancels out:
\[ \frac{\lambda_p}{\lambda_e} = \frac{c \sqrt{2mE}}{E} \]

6. **Final Expression:**
Thus, the final ratio of the wavelengths is:
\[ \frac{\lambda_p}{\lambda_e} = c \cdot \frac{\sqrt{2m}}{E} \]

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