Neglecting veriation of mass with velocity , the wavelength associated with an electron having a kinetic energy E is proportional to

Options

A. ` E^(1//2)`
B. E
C. ` E^(-1//2)`
D. `E^(-1//2)`

Answer

To solve the problem of finding the relationship between the wavelength associated with an electron and its kinetic energy \( E \), we can follow these steps:

### Step 1: Write the formula for kinetic energy
The kinetic energy \( E \) of an electron can be expressed by the formula:
\[ E = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 2: Rearrange the kinetic energy formula
We can rearrange the kinetic energy formula to express \( mv^2 \):
\[ 2E = mv^2 \]

### Step 3: Write the formula for momentum
The momentum \( p \) of the electron is given by:
\[ p = mv \]

### Step 4: Express momentum in terms of kinetic energy
From the expression for momentum, we can square both sides:
\[ p^2 = m^2v^2 \]
Now, substituting \( mv^2 \) from the kinetic energy equation:
\[ p^2 = m \cdot mv^2 = m \cdot 2E \]
This simplifies to:
\[ p^2 = 2Em \]

### Step 5: Solve for momentum
Taking the square root of both sides gives us:
\[ p = \sqrt{2Em} \]

### Step 6: Use de Broglie's wavelength formula
According to de Broglie's hypothesis, the wavelength \( \lambda \) associated with a particle is given by:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant.

### Step 7: Substitute momentum into the wavelength formula
Substituting the expression for momentum into the wavelength formula:
\[ \lambda = \frac{h}{\sqrt{2Em}} \]

### Step 8: Analyze the relationship
From the equation \( \lambda = \frac{h}{\sqrt{2Em}} \), we can see that the wavelength \( \lambda \) is inversely proportional to the square root of the kinetic energy \( E \):
\[ \lambda \propto \frac{1}{\sqrt{E}} \]

### Final Conclusion
Thus, the wavelength associated with an electron having a kinetic energy \( E \) is proportional to \( E^{-1/2} \).