The radius of `n^th` orbit `r_n` in the terms of Bohr radius `(a_0)` for a hydrogen atom is given by the relation

Options

A. `na_0`
B. `sqrt(na_0)`
C. `n^2a_0`
D. `n^3a_0`

Explanation

To find the radius of the \( n^{th} \) orbit \( r_n \) in terms of the Bohr radius \( a_0 \) for a hydrogen atom, we can follow these steps:

### Step 1: Understand the Concept of Bohr's Model
Bohr's model describes the hydrogen atom as having electrons in discrete orbits around the nucleus. Each orbit corresponds to a specific energy level.

**Hint:** Remember that in Bohr's model, the electron moves in circular orbits due to the electrostatic force between the positively charged nucleus and the negatively charged electron.

### Step 2: Identify the Forces Acting on the Electron
In the \( n^{th} \) orbit, the centripetal force required to keep the electron in circular motion is provided by the electrostatic force of attraction between the electron and the nucleus.

The electrostatic force \( F \) can be expressed as:
\[ F = \frac{k e^2}{r_n^2} \]
where \( k \) is Coulomb's constant, \( e \) is the charge of the electron, and \( r_n \) is the radius of the \( n^{th} \) orbit.

**Hint:** Recall that the centripetal force is given by \( F = \frac{m v^2}{r_n} \), where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 3: Set Up the Equation
Since the electrostatic force acts as the centripetal force, we can set these two expressions equal to each other:
\[ \frac{m v^2}{r_n} = \frac{k e^2}{r_n^2} \]

**Hint:** You will need to manipulate this equation to express \( r_n \) in terms of known constants and variables.

### Step 4: Use the Angular Momentum Quantization Condition
According to Bohr's model, the angular momentum \( L \) of the electron is quantized:
\[ L = m v r_n = \frac{n h}{2 \pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant.

**Hint:** This relationship will help you find the velocity \( v \) in terms of \( r_n \) and \( n \).

### Step 5: Solve for the Radius \( r_n \)
From the angular momentum equation, we can express \( v \) as:
\[ v = \frac{n h}{2 \pi m r_n} \]

Now, substitute this expression for \( v \) back into the centripetal force equation:
\[ \frac{m \left( \frac{n h}{2 \pi m r_n} \right)^2}{r_n} = \frac{k e^2}{r_n^2} \]

After simplifying, we can isolate \( r_n \):
\[ r_n = \frac{n^2 h^2}{4 \pi^2 m k e^2} \]

### Step 6: Define the Bohr Radius \( a_0 \)
The Bohr radius \( a_0 \) is defined as:
\[ a_0 = \frac{h^2}{4 \pi^2 m k e^2} \]

### Step 7: Express \( r_n \) in Terms of \( a_0 \)
Substituting \( a_0 \) into the equation for \( r_n \):
\[ r_n = n^2 a_0 \]

### Final Answer
Thus, the radius of the \( n^{th} \) orbit \( r_n \) in terms of the Bohr radius \( a_0 \) is given by:
\[ r_n = n^2 a_0 \]

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