Time taken by an electrons to complete one revolution in the Bohr orbit of the `H` atom is

Options

A. (a) `(4pi^(2)mr^(2))/(nh)`
B. (b) `(nh)/(4pi^(2)mr)`
C. (c) `(2pimr)/(n^(2)h^(2))`
D. (d) `h/(2pimr)`

Answer

To find the time taken by an electron to complete one revolution in the Bohr orbit of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Time Period**:
The time period (T) for an electron to complete one revolution in an orbit can be defined as the distance traveled in one complete revolution divided by the velocity of the electron.

2. **Calculate the Distance**:
The distance for one complete revolution is the circumference of the orbit, which is given by the formula:
\[ \text{Distance} = 2\pi r \]
where \( r \) is the radius of the orbit.

3. **Determine the Velocity**:
According to Bohr's second postulate, the angular momentum of the electron is quantized and is given by:
\[ L = n \frac{h}{2\pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant. The angular momentum can also be expressed as:
\[ L = mvr \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Equating the two expressions for angular momentum, we get:
\[ mvr = n \frac{h}{2\pi} \]
From this, we can solve for the velocity \( v \):
\[ v = \frac{n h}{2\pi m r} \]

4. **Substitute Velocity into the Time Period Formula**:
Now, we can substitute the expression for velocity back into the time period formula:
\[ T = \frac{2\pi r}{v} \]
Substituting \( v \):
\[ T = \frac{2\pi r}{\frac{n h}{2\pi m r}} = \frac{2\pi r \cdot 2\pi m r}{n h} \]
Simplifying this gives:
\[ T = \frac{4\pi^2 m r^2}{n h} \]

5. **Final Expression**:
Therefore, the time taken by an electron to complete one revolution in the Bohr orbit of the hydrogen atom is:
\[ T = \frac{4\pi^2 m r^2}{n h} \]