Question

The ratio of the magnitudes of electrostatic force between two protons at a distarce r apart to that between two electrons at the same distance of separation is

• A. 1:2
• B. 2:1
• C. 1:1
• D. 4:1
• E. 1:4

Answer 1

The Correct Option is C

Ratio of Electrostatic Forces: Protons vs. Electrons 

We want to determine the ratio of the magnitudes of the electrostatic force between two protons at a distance \(r\) to that between two electrons at the same distance \(r\).

Step 1: Coulomb's Law

Coulomb's Law states that the electrostatic force (F) between two point charges \(q_1\) and \(q_2\) separated by a distance \(r\) is given by:

\(F = k \frac{|q_1 q_2|}{r^2}\)

Where:

• \(F\) is the magnitude of the electrostatic force.
• \(k\) is Coulomb's constant (\(k \approx 8.9875 \times 10^9 N \cdot m^2/C^2\)).
• \(q_1\) and \(q_2\) are the magnitudes of the charges.
• \(r\) is the distance between the charges.

Step 2: Electrostatic Force between Two Protons

Let \(F_p\) be the electrostatic force between two protons. The charge of a proton is \(+e\), where \(e = 1.602 \times 10^{-19} C\).

\(F_p = k \frac{|e \cdot e|}{r^2} = k \frac{e^2}{r^2}\)

Step 3: Electrostatic Force between Two Electrons

Let \(F_e\) be the electrostatic force between two electrons. The charge of an electron is \(-e\). Note that we're looking for the *magnitude* of the force, so we use the absolute value of the product of the charges.

\(F_e = k \frac{|(-e) \cdot (-e)|}{r^2} = k \frac{e^2}{r^2}\)

Step 4: Calculate the Ratio

The ratio of the magnitudes of the electrostatic force between two protons to that between two electrons is:

\(\frac{F_p}{F_e} = \frac{k \frac{e^2}{r^2}}{k \frac{e^2}{r^2}} = 1\)

Conclusion

The ratio of the magnitudes of the electrostatic force between two protons at a distance r apart to that between two electrons at the same distance of separation is 1:1.

Answer 2

The electrostatic force between two charges is given by Coulomb's Law: \[ F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{r^2} \] In both cases: - The magnitude of charge on a proton is \( e \), and on an electron is also \( e \) - The separation distance \( r \) is the same So, the force between two protons: \[ F_p = \frac{1}{4\pi\varepsilon_0} \cdot \frac{e \cdot e}{r^2} \] Force between two electrons: \[ F_e = \frac{1}{4\pi\varepsilon_0} \cdot \frac{e \cdot e}{r^2} \] Hence, \[ \frac{F_p}{F_e} = \frac{e^2}{r^2} \div \frac{e^2}{r^2} = 1 \]

Therefore, the ratio is 1:1