A particle of charge `q` and mas `m` si accelerated from set through a potential difference `V`. Its de Broglie wavelength is equal to

Options

A. `sqrt((h)/(2mq V))`
B. `(hqV)/(sqrt(2m))`
C. `sqrt((hqV)/(2m))`
D. `(h)/(sqrt(2mqV))`

Explanation

To find the de Broglie wavelength of a particle of charge \( q \) and mass \( m \) that is accelerated through a potential difference \( V \), we can follow these steps:

### Step 1: Understand the Energy Conversion
When a charged particle is accelerated through a potential difference \( V \), the electrical potential energy is converted into kinetic energy. The potential energy gained by the particle is given by:
\[ \text{Potential Energy} = qV \]
where \( q \) is the charge of the particle and \( V \) is the potential difference.

### Step 2: Write the Kinetic Energy Equation
The kinetic energy (KE) of the particle after being accelerated is given by:
\[ \text{Kinetic Energy} = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 3: Set Up the Energy Conservation Equation
By the principle of conservation of energy, we can equate the potential energy to the kinetic energy:
\[ qV = \frac{1}{2} mv^2 \]

### Step 4: Solve for Velocity
Rearranging the equation to solve for \( v^2 \):
\[ mv^2 = 2qV \]
\[ v^2 = \frac{2qV}{m} \]

### Step 5: Find Momentum
The momentum \( p \) of the particle is given by:
\[ p = mv \]
Substituting \( v \) from the previous step:
\[ p = m \sqrt{\frac{2qV}{m}} = \sqrt{2mqV} \]

### Step 6: Use 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. Substituting the expression for momentum:
\[ \lambda = \frac{h}{\sqrt{2mqV}} \]

### Final Answer
Thus, the de Broglie wavelength of the particle is:
\[ \lambda = \frac{h}{\sqrt{2mqV}} \]

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