An electron and a photon have same wavelength `lambda`, what is ratio of their kinetic energies ?

Options

A. `h/(m_eclambda)`
B. `(2m_e clambda)/h`
C. `1:1`
D. `h/(2m_eclambda)`

Answer

To find the ratio of the kinetic energies of an electron and a photon that have the same wavelength \( \lambda \), we can follow these steps:

### Step 1: Write the expression for the kinetic energy of a photon
The kinetic energy \( E_p \) of a photon is given by the formula:
\[ E_p = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

### Step 2: Write the expression for the kinetic energy of an electron
According to de Broglie's hypothesis, the wavelength \( \lambda \) of an electron is given by:
\[ \lambda = \frac{h}{\sqrt{2m_e E_e}} \]
where \( m_e \) is the mass of the electron and \( E_e \) is the kinetic energy of the electron. Rearranging this equation gives:
\[ E_e = \frac{h^2}{2m_e \lambda^2} \]

### Step 3: Set the wavelengths equal
Since both the electron and the photon have the same wavelength \( \lambda \), we can use the expressions derived above to find the ratio of their kinetic energies.

### Step 4: Find the ratio of the kinetic energies
Now, we can find the ratio of the kinetic energy of the electron \( E_e \) to the kinetic energy of the photon \( E_p \):
\[ \frac{E_e}{E_p} = \frac{\frac{h^2}{2m_e \lambda^2}}{\frac{hc}{\lambda}} \]

### Step 5: Simplify the ratio
This simplifies to:
\[ \frac{E_e}{E_p} = \frac{h^2}{2m_e \lambda^2} \cdot \frac{\lambda}{hc} = \frac{h}{2m_e c \lambda} \]

### Step 6: Final result
Thus, the ratio of the kinetic energies of the electron and the photon is:
\[ \frac{E_e}{E_p} = \frac{h}{2m_e c \lambda} \]

### Conclusion
This gives us the final expression for the ratio of the kinetic energies of the electron and the photon.

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