If the velocity of an electron is doubled, its de-Broglie frequency will be

Options

A. half
B. remain same
C. doubled
D. become four times

Answer

To solve the problem of how the de-Broglie frequency changes when the velocity of an electron is doubled, we can follow these steps:

### 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 \(h\) is Planck's constant and \(p\) is the momentum of the particle. For an electron, the momentum \(p\) can be expressed as:
\[ p = mv \]
where \(m\) is the mass of the electron and \(v\) is its velocity.

### Step 2: Relate de-Broglie wavelength to frequency
The de-Broglie frequency (\(f\)) can be related to the wavelength and velocity by the equation:
\[ f = \frac{v}{\lambda} \]
Substituting the expression for \(\lambda\) from Step 1, we get:
\[ f = \frac{v}{\frac{h}{mv}} = \frac{mv^2}{h} \]

### Step 3: Analyze the effect of doubling the velocity
If the velocity of the electron is doubled, we can denote the new velocity as \(v' = 2v\). Now, substituting \(v'\) into the frequency equation:
\[ f' = \frac{m(v')^2}{h} = \frac{m(2v)^2}{h} = \frac{m \cdot 4v^2}{h} = 4 \cdot \frac{mv^2}{h} = 4f \]
This shows that the new frequency \(f'\) is four times the original frequency \(f\).

### Step 4: Conclusion
Thus, if the velocity of the electron is doubled, its de-Broglie frequency will be quadrupled.

### Final Answer
The de-Broglie frequency will be **4 times**.

---