Wavelength of the wave associated with a moving electron

Options

A. Decreases with increase in speed of electron
B. Increases with increase in speed of electron
C. Remains same irrespective of speed of electron
D. is zero

Answer

To find the wavelength of the wave associated with a moving electron, we can use the de Broglie wavelength formula. Here’s a step-by-step solution:

### Step 1: Understand the de Broglie Wavelength Formula
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where:
- \( \lambda \) is the wavelength,
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( p \) is the momentum of the particle.

### Step 2: Express Momentum in Terms of Mass and Velocity
The momentum (p) of an electron can be expressed as:
\[ p = m_e \cdot v_e \]
where:
- \( m_e \) is the mass of the electron (\(9.11 \times 10^{-31} \, \text{kg}\)),
- \( v_e \) is the velocity of the electron.

### Step 3: Substitute Momentum into the de Broglie Formula
Substituting the expression for momentum into the de Broglie wavelength formula gives:
\[ \lambda = \frac{h}{m_e \cdot v_e} \]

### Step 4: Analyze the Effect of Velocity on Wavelength
From the equation \( \lambda = \frac{h}{m_e \cdot v_e} \):
- As the velocity \( v_e \) increases, the denominator increases, which means that the wavelength \( \lambda \) decreases.
- Therefore, the wavelength is inversely proportional to the velocity of the electron.

### Step 5: Conclusion
Based on the analysis, we can conclude that:
- The wavelength of the wave associated with a moving electron decreases with increasing speed of the electron.

### Answer
The correct option is that the wavelength decreases with increasing speed of the electron.

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