In the Bohr's orbit, what is the ratio of total kinetic energy and total energy of the electron

Options

A. `-1`
B. `-2`
C. 1
D. `+2`

Explanation

To find the ratio of total kinetic energy (KE) to total energy (TE) of the electron in Bohr's orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Kinetic Energy (KE)**:
The kinetic energy of an electron in orbit can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

2. **Understanding Potential Energy (PE)**:
The potential energy of the electron due to electrostatic attraction can be expressed as:
\[ PE = -\frac{Ze^2}{R} \]
where \( Z \) is the atomic number, \( e \) is the charge of the electron, and \( R \) is the radius of the orbit.

3. **Using Centripetal Force**:
In Bohr's model, the centripetal force required to keep the electron in orbit is provided by the electrostatic force. Therefore, we can equate these two forces:
\[ \frac{mv^2}{R} = \frac{Ze^2}{R^2} \]
Simplifying this gives:
\[ mv^2 = \frac{Ze^2}{R} \]

4. **Relating Potential Energy to Kinetic Energy**:
We can substitute \( mv^2 \) from the previous step into the expression for potential energy:
\[ PE = -\frac{Ze^2}{R} = -2KE \]
This shows that the potential energy is twice the negative of the kinetic energy.

5. **Calculating Total Energy (TE)**:
The total energy (TE) of the electron is the sum of its kinetic and potential energies:
\[ TE = KE + PE \]
Substituting the expression for potential energy:
\[ TE = KE - 2KE = -KE \]

6. **Finding the Ratio of KE to TE**:
Now, we can find the ratio of total kinetic energy to total energy:
\[ \text{Ratio} = \frac{KE}{TE} = \frac{KE}{-KE} = -1 \]

### Final Answer:
The ratio of total kinetic energy to total energy of the electron in Bohr's orbit is:
\[ \text{Ratio} = -1 \]