The ratio of the electric force between two electrons to the gravitational force between them is of the order of

Options

A. `10^(42)`
B. `10^(40)`
C. `10^(36)`
D. `10^(32)`

Explanation

To find the ratio of the electric force between two electrons to the gravitational force between them, we can follow these steps:

### Step 1: Write the formula for electric force
The electric force \( F_e \) between two charges (in this case, two electrons) is given by Coulomb's law:
\[ F_e = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r^2} \]
where:
- \( e \) is the charge of an electron (\( e \approx 1.6 \times 10^{-19} \) C),
- \( r \) is the distance between the two electrons,
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)).

### Step 2: Write the formula for gravitational force
The gravitational force \( F_g \) between two masses (in this case, two electrons) is given by Newton's law of gravitation:
\[ F_g = \frac{G m_e^2}{r^2} \]
where:
- \( G \) is the gravitational constant (\( G \approx 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
- \( m_e \) is the mass of an electron (\( m_e \approx 9.1 \times 10^{-31} \) kg).

### Step 3: Find the ratio of electric force to gravitational force
Now, we can find the ratio \( \frac{F_e}{F_g} \):
\[ \frac{F_e}{F_g} = \frac{\frac{1}{4 \pi \epsilon_0} \frac{e^2}{r^2}}{\frac{G m_e^2}{r^2}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{e^2}{G m_e^2} \]
The \( r^2 \) terms cancel out.

### Step 4: Substitute the known values
Now we substitute the known values into the equation:
\[ \frac{F_e}{F_g} = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{(1.6 \times 10^{-19})^2}{(6.67 \times 10^{-11})(9.1 \times 10^{-31})^2} \]

### Step 5: Calculate the numerical value
Calculating the numerator:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]
Calculating the denominator:
\[ (9.1 \times 10^{-31})^2 = 8.27 \times 10^{-61} \]
Then,
\[ G m_e^2 = 6.67 \times 10^{-11} \times 8.27 \times 10^{-61} = 5.52 \times 10^{-71} \]
Now substituting these values into the ratio:
\[ \frac{F_e}{F_g} = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{2.56 \times 10^{-38}}{5.52 \times 10^{-71}} \]
Calculating the first part:
\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.11 \times 10^{-10} \]
Thus,
\[ \frac{F_e}{F_g} \approx \frac{2.56 \times 10^{-38}}{5.52 \times 10^{-71} \cdot 1.11 \times 10^{-10}} \approx \frac{2.56 \times 10^{-38}}{6.13 \times 10^{-81}} \approx 4.18 \times 10^{42} \]

### Step 6: Conclusion
The ratio of the electric force between two electrons to the gravitational force between them is of the order of \( 10^{42} \).