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

Options

A. Be halved
B. Remain same
C. Be 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 (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the electron.

### Step 2: Relate momentum to velocity
The momentum (\( p \)) of an electron can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Substituting this into the wavelength formula gives:
\[ \lambda = \frac{h}{mv} \]

### Step 3: Relate wavelength to frequency
The speed of light (\( c \)) is related to frequency (\( \nu \)) and wavelength (\( \lambda \)) by the equation:
\[ c = \nu \cdot \lambda \]
We can rearrange this to express frequency in terms of wavelength:
\[ \nu = \frac{c}{\lambda} \]

### Step 4: Substitute the expression for wavelength
Substituting the expression for \( \lambda \) from Step 2 into the frequency equation gives:
\[ \nu = \frac{c}{\frac{h}{mv}} = \frac{cmv}{h} \]
This shows that frequency (\( \nu \)) is directly proportional to the velocity (\( v \)):
\[ \nu \propto v \]

### Step 5: Analyze the effect of doubling the velocity
If the velocity of the electron is doubled (i.e., \( v \) becomes \( 2v \)), we can express the new frequency (\( \nu' \)) as:
\[ \nu' = \frac{cm(2v)}{h} = 2 \cdot \frac{cmv}{h} = 2\nu \]
This indicates that the new frequency is double the original frequency.

### Conclusion
Thus, if the velocity of the electron is doubled, its de-Broglie frequency will also be doubled.

### Final Answer
The de-Broglie frequency will be doubled.

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