The product of angular speed and tangential speed of electron in `n^"th"` orbit of hydrogen atom is

Options

A. Directly proportional to `n^2`
B. Directly proportional to `n^3`
C. Inversely proportional to `n^4`
D. independent of n

Answer

To find the product of angular speed and tangential speed of an electron in the nth orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Definitions
- **Tangential Speed (V)**: The speed of an electron moving along its circular orbit. It can be expressed as:
\[ V = \frac{2 \pi r}{T} \]
where \( r \) is the radius of the orbit and \( T \) is the time period of the electron's orbit.

- **Angular Speed (ω)**: The rate of change of angular displacement, given by:
\[ \omega = \frac{2 \pi}{T} \]

### Step 2: Relate Tangential Speed and Angular Speed
- The relationship between tangential speed and angular speed is given by:
\[ V = r \cdot \omega \]

### Step 3: Express Tangential Speed for Hydrogen Atom
For the hydrogen atom, the tangential speed \( V \) of the electron in the nth orbit can be expressed as:
\[ V = 2.18 \times 10^6 \frac{Z}{n} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 4: Express Angular Speed for Hydrogen Atom
The angular speed \( \omega \) can be expressed as:
\[ \omega = \frac{2 \pi}{T} \]
From the Bohr model, we know that the time period \( T \) is proportional to \( n^3 \):
\[ T \propto n^3 \implies \omega \propto \frac{1}{n^3} \]

### Step 5: Calculate the Product
Now, we need to find the product \( V \cdot \omega \):
\[ V \cdot \omega = \left(2.18 \times 10^6 \frac{Z}{n}\right) \cdot \left(\frac{2 \pi}{T}\right) \]
Substituting \( \omega \):
\[ V \cdot \omega \propto \frac{1}{n} \cdot \frac{1}{n^3} = \frac{1}{n^4} \]

### Conclusion
Thus, the product of angular speed and tangential speed of the electron in the nth orbit of a hydrogen atom is inversely proportional to \( n^4 \).

### Final Answer
The product of angular speed and tangential speed of the electron in the nth orbit of a hydrogen atom is:
\[ \frac{1}{n^4} \]

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