Which is the correct relation between de- Brogile wavelength of an electron in the `n^(th)` Bohr orbit and radius of the orbit R?

Options

A. `lambda = n2piR`
B. `lambda = (2piR)/n`
C. `lambda = (4 pi R)/n`
D. `lambda = (2 pi R)/nh`

Answer

To find the correct relation between the de Broglie wavelength of an electron in the \(n^{th}\) Bohr orbit and the radius of the orbit \(R\), we can follow these steps:

### Step 1: Understand de Broglie Wavelength
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: Express Momentum
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 3: Substitute Momentum into the Wavelength Formula
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{mv} \]

### Step 4: Use Bohr's Third Postulate
According to Bohr's third postulate, the angular momentum \(L\) of the electron in the \(n^{th}\) orbit is quantized and given by:
\[ L = mvr = n \frac{h}{2\pi} \]
where \(r\) is the radius of the orbit and \(n\) is a positive integer (the principal quantum number).

### Step 5: Rearrange for Velocity
From the angular momentum equation, we can express the velocity \(v\) as:
\[ v = \frac{n h}{2\pi m r} \]

### Step 6: Substitute Velocity Back into the Wavelength Formula
Now, substituting this expression for \(v\) back into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m \left(\frac{n h}{2\pi m r}\right)} = \frac{h \cdot 2\pi m r}{n h} = \frac{2\pi r}{n} \]

### Step 7: Final Relation
Thus, we arrive at the relation between the de Broglie wavelength \(\lambda\) and the radius \(r\) of the orbit:
\[ \lambda = \frac{2\pi r}{n} \]

### Conclusion
The correct relation between the de Broglie wavelength of an electron in the \(n^{th}\) Bohr orbit and the radius of the orbit \(R\) is:
\[ \lambda = \frac{2\pi R}{n} \]