The Question is from neet-physics , and it was created by the author SUBHANI

Question

Suppose the charge of a proton and an electron differ slightly. One of them is $ - e,$ the other is $(e + \Delta e).$ If the net of electrostatic force and gravitational force between two hydrogen atoms placed at a distance $d$ (much greater than atomic size) apart is zero, then $\Delta e$ is of the order of [Given mass of hydrogen ${m_h} = 1.67 	imes {10^{ - 27}}{m{kg}}$ ]

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
**Primary Concept**: Equilibrium of electrostatic and gravitational forces due to a charge difference.
**Related Topics**: Coulomb's Law, Newton's Law of Gravitation, Charge quantization
**Prerequisites**: Understanding of electrostatic force, gravitational force, and concept of charge.
## SOLUTION PATHWAY
**Given Information**:
   - Mass of hydrogen atom, $m_h = 1.67 	imes 10^{-27}$ kg
   - Charge of electron: $-e$
   - Charge of proton: $e + \Delta e$
   - Net force is zero
   - Distance between hydrogen atoms is d
**Method & Steps**:
1.  **Calculate the net charge of a hydrogen atom:**
    - A hydrogen atom has one proton and one electron.
    - Net charge of a hydrogen atom = $(e + \Delta e) + (-e) = \Delta e$
    - Since there are two hydrogen atoms, the electrostatic force between them will be due to the charge $\Delta e$ of each atom.
    - Reasoning: This step identifies the effective charge that creates the electrostatic force.
2.  **Calculate the electrostatic force:**
    - Using Coulomb's law, $F_e = k \frac{q_1 q_2}{d^2} = k \frac{(\Delta e)(\Delta e)}{d^2} = k \frac{(\Delta e)^2}{d^2}$, where $k$ is the Coulomb's constant.
    - Reasoning: This applies Coulomb's law to the given problem setup.
3.  **Calculate the gravitational force:**
    - Using Newton's law of gravitation, $F_g = G \frac{m_1 m_2}{d^2} = G \frac{m_h m_h}{d^2} = G \frac{m_h^2}{d^2}$, where $G$ is the gravitational constant.
    - Reasoning: This applies Newton's law of gravitation to the given problem setup.
4.  **Equate the magnitudes of the electrostatic and gravitational forces:**
    - Since the net force is zero, $F_e = F_g$
    - $k \frac{(\Delta e)^2}{d^2} = G \frac{m_h^2}{d^2}$
    - $(\Delta e)^2 = \frac{G m_h^2}{k}$
    - Reasoning: This sets up the equilibrium condition for solving $\Delta e$.
5. **Solve for $\Delta e$:**
    - $\Delta e = \sqrt{\frac{G m_h^2}{k}} = m_h \sqrt{\frac{G}{k}}$
    - Plugging in the values: $G = 6.67 	imes 10^{-11} Nm^2/kg^2$, $k = 9 	imes 10^9 Nm^2/C^2$, $m_h = 1.67 	imes 10^{-27}$ kg.
    - $\Delta e = 1.67 	imes 10^{-27} 	imes \sqrt{\frac{6.67 	imes 10^{-11}}{9 	imes 10^9}} \approx 1.67 	imes 10^{-27} 	imes \sqrt{0.74 	imes 10^{-20}} \approx 1.67 	imes 10^{-27} 	imes 0.86 	imes 10^{-10} \approx 1.43 	imes 10^{-37}$ C
     - Result: $\Delta e$ is of the order of $10^{-37}$ C
     - Verification: The final result is consistent with the given answer option
## OPTION ANALYSIS
[A] :x: Incorrect order of magnitude
[B] :x: Incorrect order of magnitude
[C] :x: Incorrect order of magnitude
[D] ✓: Correct order of magnitude, matches calculated value
## LEARNING AIDS
**Common Mistakes**:
1.  Error 1: Forgetting to square the charge difference when using Coulomb's law. How to avoid: Double check the formula before applying.
2.  Error 2: Using the charge of electron or proton directly instead of the net charge of the hydrogen atom. How to avoid: Understand the problem statement carefully.
**Quick Tips**:
1.  Helpful shortcut: Remember the approximate values for $k$ and $G$ to quickly estimate the solution.
2.  Memory aid: Remember that in equilibrium, the two forces must be equal in magnitude and opposite in direction.

SUBHANI · Answer

${10^{ - 47}}{\rm{C}}$

SUBHANI · Answer

${10^{ - 20}}{\rm{C}}$

SUBHANI · Answer

${10^{ - 23}}{\rm{C}}$

SUBHANI · Answer

The correct Option is "${10^{ - 37}}{\rm{C}}$"

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