Find the ratio of the magnitude of the electric force to the grativational force acting between two protons.

Explanation

To find the ratio of the magnitude of the electric force to the gravitational force acting between two protons, we can follow these steps:

### Step 1: Identify the Constants
- The charge of a proton \( q \) is \( 1.6 \times 10^{-19} \) coulombs.
- The mass of a proton \( m \) is \( 1.67 \times 10^{-27} \) kg.
- The gravitational constant \( G \) is \( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).
- The permittivity of free space \( \epsilon_0 \) is \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

### Step 2: Calculate the Electric Force
The electric force \( F_e \) between two charges is given by Coulomb's law:
\[ F_e = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 \cdot q_2}{r^2} \]
Since both charges are protons, we can substitute \( q_1 = q_2 = q \):
\[ F_e = \frac{1}{4\pi \epsilon_0} \cdot \frac{q^2}{r^2} \]
Substituting the values:
\[ F_e = \frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{r^2} \]

### Step 3: Calculate the Gravitational Force
The gravitational force \( F_g \) between two masses is given by Newton's law of gravitation:
\[ F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Since both masses are protons, we can substitute \( m_1 = m_2 = m \):
\[ F_g = \frac{G \cdot m^2}{r^2} \]
Substituting the values:
\[ F_g = \frac{6.67 \times 10^{-11} \cdot (1.67 \times 10^{-27})^2}{r^2} \]

### Step 4: Calculate the Ratio of Electric Force to Gravitational Force
Now, we can find the ratio \( \frac{F_e}{F_g} \):
\[ \frac{F_e}{F_g} = \frac{\frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{r^2}}{\frac{6.67 \times 10^{-11} \cdot (1.67 \times 10^{-27})^2}{r^2}} \]
The \( r^2 \) cancels out:
\[ \frac{F_e}{F_g} = \frac{9 \times 10^9 \cdot (1.6 \times 10^{-19})^2}{6.67 \times 10^{-11} \cdot (1.67 \times 10^{-27})^2} \]

### Step 5: Substitute the Values and Simplify
Now, substituting the values:
\[ \frac{F_e}{F_g} = \frac{9 \times 10^9 \cdot (2.56 \times 10^{-38})}{6.67 \times 10^{-11} \cdot (2.7889 \times 10^{-54})} \]
Calculating the numerator:
\[ 9 \times 10^9 \cdot 2.56 \times 10^{-38} = 2.304 \times 10^{-28} \]
Calculating the denominator:
\[ 6.67 \times 10^{-11} \cdot 2.7889 \times 10^{-54} = 1.857 \times 10^{-64} \]
Now, dividing the two results:
\[ \frac{F_e}{F_g} = \frac{2.304 \times 10^{-28}}{1.857 \times 10^{-64}} \approx 1.24 \times 10^{36} \]

### Final Answer
The ratio of the magnitude of the electric force to the gravitational force acting between two protons is approximately:
\[ \frac{F_e}{F_g} \approx 1.24 \times 10^{36} \]

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