In the Bohr's model of hydrogen atom, the radius of `n^(th)` orbit is proportional to `n^a`. Find the value of a if electric potential energy of the atom is given as :`U=U_0 ln (r/r_0)` . Here `r_0` and `U_0` are constant and r is the radius of the orbit in which electron is moving arounds the nucleus .

Explanation

To find the value of \( a \) in the equation \( r \propto n^a \) for the radius of the \( n^{th} \) orbit in the Bohr model of the hydrogen atom, given the electric potential energy \( U = U_0 \ln \left( \frac{r}{r_0} \right) \), we can follow these steps:

### Step 1: Understand the relationship between potential energy and force
The electric potential energy \( U \) is related to the force acting on the electron. The force \( F \) can be expressed as:
\[ F = -\frac{dU}{dr} \]
Calculating the derivative of \( U \) with respect to \( r \):
\[ F = -\frac{d}{dr} \left( U_0 \ln \left( \frac{r}{r_0} \right) \right) = -\frac{U_0}{r} \]

### Step 2: Relate force to centripetal force
In the Bohr model, the centripetal force required to keep the electron in circular motion is provided by the electrostatic force between the electron and the nucleus:
\[ F = \frac{mv^2}{r} = \frac{k e^2}{r^2} \]
where \( k \) is Coulomb's constant, \( e \) is the charge of the electron, and \( m \) is the mass of the electron.

### Step 3: Set the forces equal
Equating the two expressions for force:
\[ -\frac{U_0}{r} = \frac{mv^2}{r} \]
This implies:
\[ U_0 = -mv^2 \]

### Step 4: Use Bohr's quantization condition
According to Bohr's model, the angular momentum \( L \) of the electron is quantized:
\[ L = mvr = n\frac{h}{2\pi} \]
From this, we can express the velocity \( v \):
\[ v = \frac{n h}{2 \pi m r} \]

### Step 5: Substitute \( v \) into the energy equation
Substituting this expression for \( v \) into the equation \( U_0 = -mv^2 \):
\[ U_0 = -m \left( \frac{n h}{2 \pi m r} \right)^2 \]
This simplifies to:
\[ U_0 = -\frac{n^2 h^2}{4 \pi^2 r} \]

### Step 6: Rearranging for \( r \)
Rearranging the equation gives:
\[ r = -\frac{n^2 h^2}{4 \pi^2 U_0} \]
This shows that \( r \) is proportional to \( n^2 \):
\[ r \propto n^2 \]

### Step 7: Identify the value of \( a \)
From the relationship \( r \propto n^a \), we have found that \( r \propto n^2 \). Thus, we can conclude:
\[ a = 2 \]

### Final Answer
The value of \( a \) is \( 2 \).

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