The de Broglie wavelength of an electron moving in a circular orbit is `lambda` . The minimum radius of orbit is :

Options

A. `lambda/pi`
B. `lambda/(2pi)`
C. `lambda/(4pi)`
D. `lambda/(3pi)`

Explanation

To find the minimum radius of an electron moving in a circular orbit with a de Broglie wavelength of \( \lambda \), we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength \( \lambda \) 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.

### Step 2: Relate momentum to circular motion
For an electron moving in a circular orbit, the momentum \( p \) can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 3: Use the relationship between wavelength and radius
In a circular orbit, the electron's wavelength is related to the circumference of the orbit. The condition for standing waves in a circular orbit is given by:
\[ 2\pi r = n\lambda \]
where \( n \) is the principal quantum number. For the minimum radius, we take \( n = 1 \).

### Step 4: Substitute \( n = 1 \) into the equation
Substituting \( n = 1 \) into the equation gives:
\[ 2\pi r = \lambda \]

### Step 5: Solve for the minimum radius \( r \)
Rearranging the equation to solve for \( r \):
\[ r = \frac{\lambda}{2\pi} \]

### Conclusion
Thus, the minimum radius of the orbit is:
\[ r_{\text{min}} = \frac{\lambda}{2\pi} \]