The mass of an electron is `9.11 x 10^-31` kg, that of a proton is `1.67 x 10^-27` kg. Find the ratio `F_e// F_g` of the electric force and the gravitational force exerted by the proton on the electron.-

Explanation

To find the ratio \( \frac{F_e}{F_g} \) of the electric force (\( F_e \)) and the gravitational force (\( F_g \)) exerted by a proton on an electron, we can follow these steps:

### Step 1: Write the formulas for electric and gravitational forces
The gravitational force (\( F_g \)) between two masses is given by Newton's law of gravitation:
\[ F_g = \frac{G \cdot m_e \cdot m_p}{r^2} \]
where:
- \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \) (gravitational constant)
- \( m_e = 9.11 \times 10^{-31} \, \text{kg} \) (mass of the electron)
- \( m_p = 1.67 \times 10^{-27} \, \text{kg} \) (mass of the proton)
- \( r \) is the distance between the electron and proton.

The electric force (\( F_e \)) between two charges is given by Coulomb's law:
\[ F_e = \frac{k \cdot |q_e| \cdot |q_p|}{r^2} \]
where:
- \( k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \) (Coulomb's constant)
- \( q_e = -1.6 \times 10^{-19} \, \text{C} \) (charge of the electron)
- \( q_p = +1.6 \times 10^{-19} \, \text{C} \) (charge of the proton)

### Step 2: Set up the ratio \( \frac{F_e}{F_g} \)
To find the ratio of the electric force to the gravitational force, we can divide the two equations:
\[ \frac{F_e}{F_g} = \frac{\frac{k \cdot |q_e| \cdot |q_p|}{r^2}}{\frac{G \cdot m_e \cdot m_p}{r^2}} \]

### Step 3: Simplify the ratio
The \( r^2 \) terms cancel out:
\[ \frac{F_e}{F_g} = \frac{k \cdot |q_e| \cdot |q_p|}{G \cdot m_e \cdot m_p} \]

### Step 4: Substitute the known values
Now we can substitute the known values into the equation:
- \( k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)
- \( |q_e| = |q_p| = 1.6 \times 10^{-19} \, \text{C} \)
- \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( m_e = 9.11 \times 10^{-31} \, \text{kg} \)
- \( m_p = 1.67 \times 10^{-27} \, \text{kg} \)

Substituting these values gives:
\[ \frac{F_e}{F_g} = \frac{(8.99 \times 10^9) \cdot (1.6 \times 10^{-19}) \cdot (1.6 \times 10^{-19})}{(6.67 \times 10^{-11}) \cdot (9.11 \times 10^{-31}) \cdot (1.67 \times 10^{-27})} \]

### Step 5: Calculate the ratio
Calculating the numerator:
\[ 8.99 \times 10^9 \cdot (1.6 \times 10^{-19})^2 = 8.99 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.30 \times 10^{-28} \]

Calculating the denominator:
\[ 6.67 \times 10^{-11} \cdot 9.11 \times 10^{-31} \cdot 1.67 \times 10^{-27} = 6.67 \times 10^{-11} \cdot 1.51 \times 10^{-57} = 1.01 \times 10^{-67} \]

Now, substituting these values back into the ratio:
\[ \frac{F_e}{F_g} = \frac{2.30 \times 10^{-28}}{1.01 \times 10^{-67}} \approx 2.27 \times 10^{39} \]

### Final Answer
Thus, the ratio \( \frac{F_e}{F_g} \) is approximately:
\[ \frac{F_e}{F_g} \approx 2.27 \times 10^{39} \]