According to Bohr's theory, the expression for the kinetic and potential energy of an electron revolving in an orbit is given respectively by

Options

A. `+(e^(2))/(8pi epsilon_(0)r)` and `-(e^(2))/(4pi epsilon_(0)r)`
B. `+(8pi epsilon_(0)e^(2))/(r )` and `-(4pi epsilon_(0)e^(2))/(r )`
C. `-(e^(2))/(8pi epsilon_(0)r)` and `-(e^(2))/(4pi epsilon_(0)r)`
D. `+(e^(2))/(8pi epsilon_(0)r)` and `+(e^(2))/(4pi epsilon_(0)r)`

Explanation

To derive the expressions for the kinetic and potential energy of an electron revolving in an orbit according to Bohr's theory, we can follow these steps:

### Step 1: Understand the System
- Consider an electron revolving around a nucleus in a circular orbit. Let the distance between the electron and the nucleus be denoted as \( r \).
- The electron has a charge of \( -e \) and the nucleus (for a hydrogen atom, for example) has a charge of \( +e \).

### Step 2: Calculate the Potential Energy (PE)
- The potential energy \( U \) of the electron in the electric field of the nucleus is given by the formula:
\[ U = -\frac{k \cdot e^2}{r} \]
where \( k \) is Coulomb's constant.

### Step 3: Relate Kinetic Energy (KE) to Potential Energy
- According to classical mechanics, the kinetic energy \( K \) of the electron is related to the potential energy by the equation:
\[ K = -\frac{1}{2} U \]
- Substituting the expression for potential energy into this equation gives:
\[ K = -\frac{1}{2} \left(-\frac{k \cdot e^2}{r}\right) = \frac{k \cdot e^2}{2r} \]

### Step 4: Expressing in Terms of Permittivity
- We can express \( k \) in terms of the permittivity of free space \( \epsilon_0 \):
\[ k = \frac{1}{4 \pi \epsilon_0} \]
- Substituting this into the kinetic energy expression:
\[ K = \frac{1}{2} \cdot \frac{e^2}{4 \pi \epsilon_0 r} = \frac{e^2}{8 \pi \epsilon_0 r} \]

### Step 5: Final Expressions
- Thus, the final expressions for the kinetic and potential energy of the electron are:
- Potential Energy:
\[ U = -\frac{k \cdot e^2}{r} = -\frac{e^2}{4 \pi \epsilon_0 r} \]
- Kinetic Energy:
\[ K = \frac{e^2}{8 \pi \epsilon_0 r} \]

### Summary
- The potential energy of the electron is given by:
\[ U = -\frac{e^2}{4 \pi \epsilon_0 r} \]
- The kinetic energy of the electron is given by:
\[ K = \frac{e^2}{8 \pi \epsilon_0 r} \]