Two identical satellites are at R and 7R away from earth surface the wrong statement is (R = Radius of earth)

Options

A. Rate of total energy will be 4
B. Ratio of kinetic energies will be 4
C. Ratio of potential energies will be 4
D. Ratio of total energy will be 4 but ratio of potential and kinetic energies will be 2

Explanation

To solve the problem, we need to analyze the energies of two identical satellites located at distances R and 7R from the Earth's surface. Here, R is the radius of the Earth.

### Step-by-Step Solution:

1. **Identify Distances from the Center of the Earth:**
- The distance of the first satellite (Satellite 1) from the center of the Earth is:
\[ d_1 = R + R = 2R \]
- The distance of the second satellite (Satellite 2) from the center of the Earth is:
\[ d_2 = R + 7R = 8R \]

2. **Calculate Kinetic Energy (KE) of Each Satellite:**
- The formula for the kinetic energy of a satellite in orbit is given by:
\[ KE = -\frac{GMm}{2d} \]
- For Satellite 1:
\[ KE_1 = -\frac{GMm}{2 \times 2R} = -\frac{GMm}{4R} \]
- For Satellite 2:
\[ KE_2 = -\frac{GMm}{2 \times 8R} = -\frac{GMm}{16R} \]

3. **Calculate the Ratio of Kinetic Energies:**
- The ratio of kinetic energies is:
\[ \text{Ratio of KE} = \frac{KE_1}{KE_2} = \frac{-\frac{GMm}{4R}}{-\frac{GMm}{16R}} = \frac{16}{4} = 4 \]

4. **Calculate Total Energy (TE) of Each Satellite:**
- The total energy (TE) of a satellite is given by:
\[ TE = KE + PE \]
- The potential energy (PE) is given by:
\[ PE = -\frac{GMm}{d} \]
- For Satellite 1:
\[ PE_1 = -\frac{GMm}{2R} \]
\[ TE_1 = KE_1 + PE_1 = -\frac{GMm}{4R} - \frac{GMm}{2R} = -\frac{GMm}{4R} - \frac{2GMm}{4R} = -\frac{3GMm}{4R} \]
- For Satellite 2:
\[ PE_2 = -\frac{GMm}{8R} \]
\[ TE_2 = KE_2 + PE_2 = -\frac{GMm}{16R} - \frac{GMm}{8R} = -\frac{GMm}{16R} - \frac{2GMm}{16R} = -\frac{3GMm}{16R} \]

5. **Calculate the Ratio of Total Energies:**
- The ratio of total energies is:
\[ \text{Ratio of TE} = \frac{TE_1}{TE_2} = \frac{-\frac{3GMm}{4R}}{-\frac{3GMm}{16R}} = \frac{16}{4} = 4 \]

6. **Calculate Potential Energy (PE) of Each Satellite:**
- For Satellite 1:
\[ PE_1 = -\frac{GMm}{2R} \]
- For Satellite 2:
\[ PE_2 = -\frac{GMm}{8R} \]

7. **Calculate the Ratio of Potential Energies:**
- The ratio of potential energies is:
\[ \text{Ratio of PE} = \frac{PE_1}{PE_2} = \frac{-\frac{GMm}{2R}}{-\frac{GMm}{8R}} = \frac{8}{2} = 4 \]

### Conclusion:
- The ratios of kinetic energy, total energy, and potential energy are all equal to 4.
- The wrong statement among the options provided is the one that states that the ratio of potential and kinetic energy is 2.