The electron in a hydrogen atom jumps from ground state to the higher energy state where its velcoity is reduced to one-third its initial value. If the radius of the orbit in the ground state is `r` the radius of new orbit will be

Options

A. 3 r
B. 9 r
C. `( r)/(3)`
D. `( r)/(9)`

Explanation

To solve the problem, we need to analyze the relationship between the velocity of the electron, the principal quantum number \( n \), and the radius of the orbit in a hydrogen atom.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The electron is in the ground state of the hydrogen atom. In this state, the principal quantum number \( n = 1 \).
- Let the initial velocity of the electron be \( v \).
- The radius of the orbit in the ground state is given as \( r \).

2. **Determine Final Conditions**:
- The electron jumps to a higher energy state where its velocity is reduced to one-third of its initial value. Thus, the final velocity \( v' = \frac{v}{3} \).

3. **Relate Velocity and Principal Quantum Number**:
- The velocity of the electron in a hydrogen atom can be expressed as:
\[ v = \frac{Z e^2}{2 n \hbar \epsilon_0} \]
For hydrogen, \( Z = 1 \), so:
\[ v = \frac{e^2}{2 n \hbar \epsilon_0} \]
- Since \( v \) is inversely proportional to \( n \), we can write:
\[ v \propto \frac{1}{n} \]

4. **Set Up the Proportionality for the New State**:
- Let the new principal quantum number after the jump be \( n' \). Since the velocity is reduced to \( \frac{v}{3} \), we have:
\[ \frac{v}{3} \propto \frac{1}{n'} \]
- From the initial condition, we have:
\[ v \propto \frac{1}{n} \]
- Thus, we can relate the two states:
\[ \frac{v}{3} = \frac{v}{n} \implies n' = 3n \]
- Since \( n = 1 \) in the ground state, we find:
\[ n' = 3 \]

5. **Relate Radius and Principal Quantum Number**:
- The radius of the orbit in a hydrogen atom is given by:
\[ r \propto n^2 \]
- Therefore, the radius in the new state \( r' \) can be expressed as:
\[ r' \propto (n')^2 = (3)^2 = 9 \]
- Since the radius in the ground state is \( r \), we have:
\[ r' = 9r \]

### Final Answer:
The radius of the new orbit will be \( 9r \).