To determine the ratio of the electrostatic force \( F_E \) to the gravitational force \( F_G \) between two protons, we need to start by recalling the formulas for both forces. ### Step 1: Formulas for Electrostatic and Gravitational Forces 1. **Electrostatic Force (\( F_E \))**: The force between two charged particles is given by Coulomb's Law: \[ F_E = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \] where: - \( k = \frac{1}{4\pi \epsilon_0} \) is Coulomb's constant, - \( q_1 \) and \( q_2 \) are the charges of the particles (for protons, \( q_1 = q_2 = e \)), - \( r \) is the distance between the charges (in this case, \( d \)). Thus, for two protons: \[ F_E = \frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{d^2} \] 2. **Gravitational Force (\( F_G \))**: The force between two masses is given by Newton's Law of Gravitation: \[ F_G = \frac{G \cdot m_1 \cdot m_2}{r^2} \] where: - \( G \) is the gravitational constant, - \( m_1 \) and \( m_2 \) are the masses of the objects (for protons, \( m_1 = m_2 = m \)), - \( r \) is the distance between the masses (again, \( d \)). Thus, for two protons: \[ F_G = \frac{G \cdot m^2}{d^2} \] ### Step 2: Ratio of Electrostatic Force to Gravitational Force Now, we can find the ratio \( \frac{F_E}{F_G} \): \[ \frac{F_E}{F_G} = \frac{\frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{d^2}}{\frac{G \cdot m^2}{d^2}} \] ### Step 3: Simplifying the Ratio Notice that \( d^2 \) cancels out: \[ \frac{F_E}{F_G} = \frac{\frac{1}{4\pi \epsilon_0} \cdot e^2}{G \cdot m^2} \] This can be rewritten as: \[ \frac{F_E}{F_G} = \frac{e^2}{4\pi \epsilon_0 G m^2} \] ### Step 4: Identifying the Correct Option Now, looking at the options provided: - A. \(\frac{e}{4\pi \epsilon_0 G m}\) - B. \(\frac{e^2}{G m^2}\) - C. \(\frac{G m^2}{4\pi \epsilon_0 e^2}\) - D. \(\frac{e^2}{4\pi \epsilon_0 G m^2}\) The correct answer is **D**: \(\frac{e^2}{4\pi \epsilon_0 G m^2}\). ### Step 5: Explanation of Incorrect Options - **Option A**: \(\frac{e}{4\pi \epsilon_0 G m}\) is incorrect because it has the charge \( e \) instead of \( e^2 \) and the mass \( m \) instead of \( m^2 \). The dimensions do not match the forces being compared. - **Option B**: \(\frac{e^2}{G m^2}\) is incorrect because it lacks the \( 4\pi \epsilon_0 \) factor in the denominator, which is essential for the electrostatic force calculation. - **Option C**: \(\frac{G m^2}{4\pi \epsilon_0 e^2}\) is incorrect because it has the terms inverted. It suggests a ratio that does not represent the relationship between the forces correctly. ### Summary - The ratio of electrostatic force to gravitational force between two protons is given by \(\frac{e^2}{4\pi \epsilon_0 G m^2}\). - The electrostatic force is significantly stronger than the gravitational force due to the nature of the forces involved. - Always ensure to use the correct formulas for both electrostatic and gravitational forces when calculating ratios. - Pay attention to the units and dimensions of the quantities involved to avoid common pitfalls in physics problems. This thorough understanding will help you tackle similar problems in your exams effectively!