A particle moving with kinetic energy `E_(1)` has de Broglie wavelength `lambda_(1)`. Another particle of same mass having kinetic energy `E_(2)` has de Broglie wavelength `lambda_(2). lambda_(2)= 3 lambda_(2)`. Then `E_(2) - E_(1)` is equal to

Options

A. `7E_(1)`
B. `3E_(1)`
C. `8E_(1)`
D. `9E_(1)`

Explanation

To solve the problem, we need to relate the kinetic energy of the particles to their de Broglie wavelengths. Let's break it down step by step.

### Step 1: Write the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is the Planck constant and \( p \) is the momentum of the particle.

### Step 2: Express momentum in terms of kinetic energy
The momentum (\( p \)) of a particle can also be expressed in terms of its kinetic energy (\( E \)):
\[ E = \frac{p^2}{2m} \implies p = \sqrt{2mE} \]
where \( m \) is the mass of the particle.

### Step 3: Substitute momentum into the de Broglie wavelength formula
Substituting \( p \) into the de Broglie wavelength formula gives:
\[ \lambda = \frac{h}{\sqrt{2mE}} \]

### Step 4: Relate kinetic energy to de Broglie wavelength
From the above equation, we can express kinetic energy in terms of wavelength:
\[ E = \frac{h^2}{2m\lambda^2} \]
This shows that kinetic energy is inversely related to the square of the wavelength.

### Step 5: Set up the relationship between the two particles
Let \( E_1 \) and \( E_2 \) be the kinetic energies of the two particles with wavelengths \( \lambda_1 \) and \( \lambda_2 \), respectively. Given that \( \lambda_2 = 3\lambda_1 \), we can write:
\[ E_1 = \frac{h^2}{2m\lambda_1^2} \]
\[ E_2 = \frac{h^2}{2m\lambda_2^2} = \frac{h^2}{2m(3\lambda_1)^2} = \frac{h^2}{2m \cdot 9\lambda_1^2} = \frac{1}{9} \cdot \frac{h^2}{2m\lambda_1^2} = \frac{1}{9} E_1 \]

### Step 6: Calculate \( E_2 \) in terms of \( E_1 \)
Since we have established that:
\[ E_2 = \frac{1}{9} E_1 \]

### Step 7: Find \( E_2 - E_1 \)
Now, we need to find \( E_2 - E_1 \):
\[ E_2 - E_1 = \frac{1}{9} E_1 - E_1 = \left(\frac{1}{9} - 1\right) E_1 = \left(-\frac{8}{9}\right) E_1 \]

### Final Answer
Thus, the difference in kinetic energy \( E_2 - E_1 \) is:
\[ E_2 - E_1 = -\frac{8}{9} E_1 \]