An artificial satellite moves in a circular orbit around the earth. Total energy of the satellite is E. Then what is the potential energy of satellite

Options

A. `- 2E`
B. `2E`
C. `(2 E)/(3)`
D. `(- 2 E)/(3)`

Explanation

To solve the problem, we need to derive the potential energy of an artificial satellite moving in a circular orbit around the Earth, given its total energy \( E \).

### Step-by-Step Solution:

1. **Understanding Total Energy of the Satellite**:
The total energy \( E \) of a satellite in a circular orbit is given by the formula:
\[ E = K + U \]
where \( K \) is the kinetic energy and \( U \) is the potential energy.

2. **Kinetic Energy of the Satellite**:
The kinetic energy \( K \) of a satellite in orbit can be expressed as:
\[ K = \frac{GMm}{2R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( R \) is the distance from the center of the Earth to the satellite.

3. **Potential Energy of the Satellite**:
The potential energy \( U \) of the satellite is given by:
\[ U = -\frac{GMm}{R} \]

4. **Relating Total Energy and Potential Energy**:
We know that the total energy \( E \) can also be expressed in terms of potential energy:
\[ E = K + U \]
Substituting the expressions for \( K \) and \( U \):
\[ E = \frac{GMm}{2R} - \frac{GMm}{R} \]

5. **Simplifying the Expression**:
To combine the terms, note that:
\[ -\frac{GMm}{R} = -2 \cdot \frac{GMm}{2R} \]
Therefore, we can rewrite the total energy as:
\[ E = \frac{GMm}{2R} - 2 \cdot \frac{GMm}{2R} = -\frac{GMm}{2R} \]

6. **Finding Potential Energy**:
From the expression for total energy, we can rearrange to find the potential energy:
\[ U = 2E \]
Thus, the potential energy of the satellite is:
\[ U = -\frac{GMm}{R} = 2E \]

### Final Answer:
The potential energy of the satellite is given by:
\[ U = 2E \]