The relationship between kinetic energy (K) and potential energy (U) of electron moving in a orbit around the nucleus is

Options

A. `U=-K`
B. `U=-2K`
C. `U=-3K`
D. `U=-(1)/(2)K`

Explanation

To find the relationship between the kinetic energy (K) and potential energy (U) of an electron moving in an orbit around the nucleus, we can follow these steps:

### Step 1: Understand the Forces Acting on the Electron
The electron revolves around the nucleus due to the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron. This force acts as the centripetal force required for circular motion.

### Step 2: Write the Expression for Centripetal Force
The centripetal force required to keep the electron in circular motion can be expressed as:
\[ F_{\text{centripetal}} = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron,
- \( r \) is the radius of the orbit.

### Step 3: Write the Expression for Electrostatic Force
The electrostatic force between the nucleus and the electron can be expressed using Coulomb's law:
\[ F_{\text{electrostatic}} = \frac{kZe^2}{r^2} \]
where:
- \( k \) is Coulomb's constant,
- \( Z \) is the atomic number (number of protons in the nucleus),
- \( e \) is the charge of the electron.

### Step 4: Set the Forces Equal
Since the electrostatic force provides the necessary centripetal force, we can set these two forces equal to each other:
\[ \frac{kZe^2}{r^2} = \frac{mv^2}{r} \]

### Step 5: Simplify the Equation
By multiplying both sides by \( r \) and rearranging, we get:
\[ mv^2 = \frac{kZe^2}{r} \]

### Step 6: Find the Kinetic Energy (K)
The kinetic energy (K) of the electron can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
Substituting \( mv^2 \) from the previous step:
\[ K = \frac{1}{2} \left(\frac{kZe^2}{r}\right) = \frac{kZe^2}{2r} \]

### Step 7: Find the Potential Energy (U)
The potential energy (U) of the electron in the electric field of the nucleus is given by:
\[ U = -\frac{kZe^2}{r} \]

### Step 8: Relate Kinetic Energy and Potential Energy
From the expressions for K and U, we can relate them:
\[ U = -2K \]
This shows that the potential energy is twice the negative of the kinetic energy.

### Final Result
Thus, the relationship between kinetic energy (K) and potential energy (U) of an electron moving in an orbit around the nucleus is:
\[ U = -2K \]