If `E_(1) , E_(2) " and " E_(3)` represent respectively the kinetic energies of an electron , an alpha particle and a proton each having same de Broglie wavelength then :

Options

A. `E_(1) gt E_(3) gt E_(2)`
B. `E_(2) gt E_(3) gt E_(1)`
C. `E_(1) gt E_(2) gt E_(3)`
D. `E_(1) = E_(2) = E_(3)`

Answer

To solve the problem, we need to determine the relationship between the kinetic energies of an electron, an alpha particle, and a proton, all having the same de Broglie wavelength. Let's break it down step by step.

### Step 1: Understanding de Broglie Wavelength
The de Broglie wavelength (\( \lambda \)) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 2: Expressing Velocity in Terms of Wavelength
From the de Broglie wavelength formula, we can express the velocity \( v \) as:
\[ v = \frac{h}{m\lambda} \]

### Step 3: Kinetic Energy Formula
The kinetic energy (\( E \)) of a particle is given by:
\[ E = \frac{1}{2} mv^2 \]
Substituting the expression for \( v \) into the kinetic energy formula, we get:
\[ E = \frac{1}{2} m \left(\frac{h}{m\lambda}\right)^2 \]

### Step 4: Simplifying the Kinetic Energy Expression
Now, simplifying the equation:
\[ E = \frac{1}{2} m \cdot \frac{h^2}{m^2 \lambda^2} \]
This simplifies to:
\[ E = \frac{h^2}{2m\lambda^2} \]

### Step 5: Analyzing the Kinetic Energy for Different Particles
Since \( \lambda \) is the same for all three particles (electron, proton, and alpha particle), we can see that the kinetic energy is inversely proportional to the mass of the particle:
\[ E \propto \frac{1}{m} \]
This means that the smaller the mass, the larger the kinetic energy.

### Step 6: Comparing Masses of the Particles
- Mass of electron (\( m_e \)) is much smaller than the mass of proton (\( m_p \)).
- Mass of alpha particle (\( m_{\alpha} \)) is approximately \( 2m_p \).

### Step 7: Ordering the Kinetic Energies
From the relationship \( E \propto \frac{1}{m} \):
- The electron has the highest kinetic energy because it has the smallest mass.
- The proton has a higher kinetic energy than the alpha particle because it has a smaller mass than the alpha particle.
- The alpha particle has the lowest kinetic energy because it has the largest mass.

Thus, the order of kinetic energies is:
\[ E_1 > E_3 > E_2 \]
where:
- \( E_1 \) is the kinetic energy of the electron,
- \( E_3 \) is the kinetic energy of the proton,
- \( E_2 \) is the kinetic energy of the alpha particle.

### Final Answer
The order of kinetic energies is:
\[ E_1 > E_3 > E_2 \]