If the velocity of a moving particle is reduced to half, then percentage change in its wavelength will be

Options

A. 100% decrease
B. 100% increase
C. 50% decrease
D. 50% increase

Explanation

To solve the problem of how the wavelength of a moving particle changes when its velocity is reduced to half, we can follow these steps:

### Step 1: Understand the formula for 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,
- \( v \) is the velocity of the particle.

### Step 2: Calculate the initial wavelength
Let the initial velocity of the particle be \( v \). The initial wavelength (\( \lambda_1 \)) can be expressed as:
\[ \lambda_1 = \frac{h}{mv} \]

### Step 3: Calculate the new wavelength after reducing the velocity
If the velocity is reduced to half, the new velocity becomes \( \frac{v}{2} \). The new wavelength (\( \lambda_2 \)) can be calculated as:
\[ \lambda_2 = \frac{h}{m \cdot \frac{v}{2}} = \frac{h \cdot 2}{mv} = 2 \cdot \frac{h}{mv} = 2\lambda_1 \]

### Step 4: Determine the change in wavelength
Now we can find the change in wavelength:
\[ \Delta \lambda = \lambda_2 - \lambda_1 = 2\lambda_1 - \lambda_1 = \lambda_1 \]

### Step 5: Calculate the percentage change in wavelength
The percentage change in wavelength can be calculated using the formula:
\[ \text{Percentage Change} = \frac{\Delta \lambda}{\lambda_1} \times 100\% \]
Substituting the values we found:
\[ \text{Percentage Change} = \frac{\lambda_1}{\lambda_1} \times 100\% = 100\% \]

### Conclusion
Thus, when the velocity of a moving particle is reduced to half, the percentage change in its wavelength is **100%**.

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