If e is the charge of an electron in esu m is the mass in grams and v the voltage 'h' is the planck constant in erg sec then the wavelength of the electron in cm is

Options

A. `(h)/sqrt(Ve)`
B. `(h)/sqrt(Vme)`
C. `sqrt(h)/(ve)`
D. `(h)/sqrt(2mev)`

Answer

To find the wavelength of an electron in centimeters, we can use the de Broglie wavelength formula, which relates the wavelength (λ) of a particle to its momentum (p). The formula is given by:

\[ \lambda = \frac{h}{p} \]

Where:
- \( h \) is the Planck's constant in erg seconds.
- \( p \) is the momentum of the electron.

### Step 1: Express momentum in terms of mass and velocity
The momentum (p) of an electron can be expressed as:

\[ p = mv \]

Where:
- \( m \) is the mass of the electron in grams.
- \( v \) is the velocity of the electron.

### Step 2: Relate kinetic energy to velocity
The kinetic energy (KE) of the electron can be expressed as:

\[ KE = \frac{1}{2} mv^2 \]

We can also express kinetic energy in terms of the charge (e) of the electron and the voltage (V):

\[ KE = eV \]

### Step 3: Solve for velocity
From the kinetic energy equation, we can solve for \( v \):

\[ \frac{1}{2} mv^2 = eV \implies mv^2 = 2eV \implies v^2 = \frac{2eV}{m} \implies v = \sqrt{\frac{2eV}{m}} \]

### Step 4: Substitute momentum into the de Broglie equation
Now, substituting \( v \) back into the momentum equation:

\[ p = mv = m\sqrt{\frac{2eV}{m}} = \sqrt{2meV} \]

### Step 5: Substitute momentum into the de Broglie wavelength equation
Now we can substitute this expression for momentum into the de Broglie wavelength formula:

\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2meV}} \]

### Step 6: Convert wavelength to centimeters
Since the wavelength is required in centimeters, we need to ensure that \( h \) is in erg seconds, \( m \) is in grams, and \( e \) and \( V \) are in appropriate units. The final expression for the wavelength in centimeters is:

\[ \lambda = \frac{h}{\sqrt{2meV}} \text{ (in cm)} \]

### Final Answer
Thus, the wavelength of the electron in centimeters is given by:

\[ \lambda = \frac{h}{\sqrt{2meV}} \]