The de Broglie wavelength relates to applied voltage as:

Options

A. `lambda = (12.3)/(sqrt(h))overset(0)(A)`
B. `lambda =(12.3)/(sqrt(V))overset(0)(A)`
C. `lambda = (12.3)/(sqrt(E))overset(0)(A)`
D. Both (2) and (3)

Explanation

To derive the relationship between the de Broglie wavelength and the applied voltage, we will follow these steps:

### Step 1: Understand the Kinetic Energy of an Electron
The kinetic energy (KE) of an electron accelerated through a potential difference (V) is given by:
\[ KE = eV \]
where \( e \) is the charge of the electron and \( V \) is the applied voltage.

### Step 2: Relate Kinetic Energy to Velocity
The kinetic energy can also be expressed in terms of the mass (m) and velocity (v) of the electron:
\[ KE = \frac{1}{2} mv^2 \]
Setting the two expressions for kinetic energy equal gives:
\[ eV = \frac{1}{2} mv^2 \]

### Step 3: Solve for Velocity
From the equation \( eV = \frac{1}{2} mv^2 \), we can solve for \( v \):
\[ mv^2 = 2eV \]
\[ v^2 = \frac{2eV}{m} \]
\[ v = \sqrt{\frac{2eV}{m}} \]

### Step 4: Write the de Broglie Wavelength Formula
The de Broglie wavelength (\( \lambda \)) is given by:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant.

### Step 5: Substitute Velocity into the Wavelength Formula
Substituting the expression for \( v \) from Step 3 into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m\sqrt{\frac{2eV}{m}}} \]
This simplifies to:
\[ \lambda = \frac{h}{\sqrt{2emV}} \]

### Step 6: Final Expression for de Broglie Wavelength
Thus, the final expression relating the de Broglie wavelength to the applied voltage is:
\[ \lambda = \frac{h}{\sqrt{2emV}} \]

### Conclusion
This equation shows that the de Broglie wavelength is inversely proportional to the square root of the applied voltage \( V \).