de-Broglie wavelength of an electron in the nth Bohr orbit is `lambda_(n)` and the angular momentum is `J_(n)` then

Options

A. `J_(n)proplambda_(n)`
B. `lambda_(n)prop(1)/(J_(n))`
C. `lambda_(n)propJ_(n)^(2)`
D. None of these

Answer

To find the relationship between the de Broglie wavelength \( \lambda_n \) of an electron in the nth Bohr orbit and its angular momentum \( J_n \), we can follow these steps:

### Step 1: Write the expression for de Broglie wavelength
The de Broglie wavelength \( \lambda_n \) is given by the formula:
\[ \lambda_n = \frac{h}{P_n} \]
where \( h \) is Planck's constant and \( P_n \) is the linear momentum of the electron in the nth orbit.

### Step 2: Relate linear momentum to angular momentum
The angular momentum \( J_n \) of the electron in the nth orbit can be expressed as:
\[ J_n = mvr \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the orbit.

We can also express linear momentum \( P_n \) as:
\[ P_n = mv \]
Thus, we can relate the angular momentum to linear momentum:
\[ J_n = P_n \cdot r \]
From this, we can express \( P_n \) in terms of \( J_n \):
\[ P_n = \frac{J_n}{r} \]

### Step 3: Substitute \( P_n \) into the de Broglie wavelength formula
Now, we can substitute \( P_n \) back into the expression for \( \lambda_n \):
\[ \lambda_n = \frac{h}{P_n} = \frac{h}{\frac{J_n}{r}} = \frac{hr}{J_n} \]

### Step 4: Analyze the relationship
From the equation \( \lambda_n = \frac{hr}{J_n} \), we can see that:
\[ \lambda_n \propto \frac{1}{J_n} \]
This indicates that the de Broglie wavelength \( \lambda_n \) is inversely proportional to the angular momentum \( J_n \).

### Conclusion
Thus, the relationship between the de Broglie wavelength and the angular momentum is:
\[ \lambda_n \text{ is inversely proportional to } J_n \]

### Final Answer
The correct option is that \( \lambda_n \) is inversely proportional to \( J_n \).

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