Suppose the charge of a proton and an electron differ slightly. One of them is `-e`, the other is `(e+Deltae)`. 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 `Deltae` is of the order of [Given mass of hydrogen `m_(h)=1.67xx10^(-27)kg`]

Options

A. `10^(-23) C`
B. `10^(-37) C`
C. `10^(-47) C`
D. `10^(-20) C`

Explanation

To solve the problem, we need to find the order of \(\Delta e\) given that the net electrostatic force and gravitational force between two hydrogen atoms is zero. We will start by equating the electrostatic force and gravitational force.

### Step 1: Write the expressions for the forces
1. **Electrostatic Force (\(F_e\))**: The electrostatic force between two charges is given by Coulomb's law:
\[ F_e = k \frac{(e + \Delta e)(-e)}{d^2} \]
Here, \(k\) is Coulomb's constant, and \(d\) is the distance between the two hydrogen atoms.

2. **Gravitational Force (\(F_g\))**: The gravitational force between two masses is given by Newton's law of gravitation:
\[ F_g = G \frac{m_h^2}{d^2} \]
where \(G\) is the gravitational constant and \(m_h\) is the mass of a hydrogen atom.

### Step 2: Set the forces equal to each other
Since the net force is zero, we can set the magnitudes of the electrostatic force and gravitational force equal to each other:
\[ F_e = F_g \]
Substituting the expressions we have:
\[ k \frac{(e + \Delta e)(-e)}{d^2} = G \frac{m_h^2}{d^2} \]

### Step 3: Simplify the equation
We can cancel \(d^2\) from both sides:
\[ k (e + \Delta e)(-e) = G m_h^2 \]
This simplifies to:
\[ -k e (e + \Delta e) = G m_h^2 \]

### Step 4: Expand and rearrange
Expanding the left side:
\[ -k e^2 - k e \Delta e = G m_h^2 \]
Rearranging gives:
\[ -k e \Delta e = G m_h^2 + k e^2 \]

### Step 5: Solve for \(\Delta e\)
Now, we can isolate \(\Delta e\):
\[ \Delta e = -\frac{G m_h^2 + k e^2}{k e} \]
For small \(\Delta e\), we can neglect the term \(G m_h^2\) compared to \(k e^2\) since \(e\) is much larger than \(\Delta e\):
\[ \Delta e \approx -\frac{k e^2}{k e} = -e \]

### Step 6: Calculate the order of \(\Delta e\)
To find the order of \(\Delta e\), we can use the known values:
- \(k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2\)
- \(G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)
- \(m_h = 1.67 \times 10^{-27} \, \text{kg}\)
- \(e \approx 1.6 \times 10^{-19} \, \text{C}\)

Substituting these values into the equation for \(\Delta e\):
\[ \Delta e \approx \sqrt{\frac{G m_h^2}{k}} \]

Calculating the order:
1. Calculate \(G m_h^2\):
\[ G m_h^2 = 6.67 \times 10^{-11} \times (1.67 \times 10^{-27})^2 \approx 6.67 \times 10^{-11} \times 2.7889 \times 10^{-54} \approx 1.86 \times 10^{-64} \]

2. Calculate \(\Delta e\):
\[ \Delta e \approx \sqrt{\frac{1.86 \times 10^{-64}}{9 \times 10^9}} \approx \sqrt{2.0667 \times 10^{-74}} \approx 1.44 \times 10^{-37} \]

### Final Result
Thus, \(\Delta e\) is of the order of \(10^{-37} \, \text{C}\).

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