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

Options

A. `(4pi^(2)mr^(2))/(hm)`
B. `(nh)/(4pi^(2)mr)`
C. `(2pimr)/(n^(2)h^(2))`
D. `(h)/(2pimr)`

Explanation

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 1: Understand the Concept
The time taken for one complete revolution in a circular orbit can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

### Step 2: Determine the Distance
In a circular orbit, the distance traveled in one complete revolution is the circumference of the circle. The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the orbit.

### Step 3: Determine the Speed
The speed \( v \) of the electron in the orbit can be derived from Bohr's postulate, which states:
\[ mvr = \frac{nh}{2\pi} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron,
- \( r \) is the radius of the orbit,
- \( n \) is the principal quantum number,
- \( h \) is Planck's constant.

From this equation, we can express the speed \( v \) as:
\[ v = \frac{nh}{2\pi m r} \]

### Step 4: Substitute into the Time Formula
Now we can substitute the expressions for distance and speed into the time formula:
\[ \text{Time} = \frac{C}{v} = \frac{2\pi r}{\frac{nh}{2\pi m r}} \]

### Step 5: Simplify the Expression
Simplifying the above expression:
\[ \text{Time} = \frac{2\pi r \cdot 2\pi m r}{nh} \]
\[ \text{Time} = \frac{4\pi^2 m r^2}{nh} \]

### Final Answer
Thus, the time taken by an electron to complete one revolution in the Bohr orbit of the hydrogen atom is:
\[ \text{Time} = \frac{4\pi^2 m r^2}{nh} \]