At what distance should two charges, each equals to 1C, be placed so that the force between them equals your weight?

Explanation

To solve the problem of finding the distance at which two charges of 1C each should be placed so that the electrostatic force between them equals your weight, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let \( q_1 = q_2 = 1 \, \text{C} \) (the charges).
- Let \( F \) be the electrostatic force between the charges.
- Let \( m \) be your weight in kilograms.
- The acceleration due to gravity \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

2. **Write the Formula for Electrostatic Force:**
The electrostatic force \( F \) between two point charges is given by Coulomb's Law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
where \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

3. **Set the Electrostatic Force Equal to Weight:**
Your weight \( W \) is given by:
\[ W = m \cdot g \]
We want to find the distance \( r \) such that:
\[ F = W \]

4. **Substitute the Values:**
Substituting the values of \( q_1 \) and \( q_2 \) into the force equation:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{(1)(1)}{r^2} \]
Therefore, we have:
\[ \frac{1}{4 \pi \epsilon_0} \frac{1}{r^2} = m \cdot g \]

5. **Rearranging the Equation:**
Rearranging gives:
\[ r^2 = \frac{1}{4 \pi \epsilon_0 (m \cdot g)} \]
Taking the square root:
\[ r = \sqrt{\frac{1}{4 \pi \epsilon_0 (m \cdot g)}} \]

6. **Substituting Known Values:**
Using \( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \), \( g \approx 9.8 \, \text{m/s}^2 \), and substituting a weight \( m \) (for example, \( m = 64 \, \text{kg} \)):
\[ r = \sqrt{\frac{1}{4 \pi (8.85 \times 10^{-12}) (64 \cdot 9.8)}} \]

7. **Calculating the Distance:**
- Calculate \( 4 \pi \epsilon_0 \):
\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.112 \times 10^{-10} \]
- Calculate \( 64 \cdot 9.8 \):
\[ 64 \cdot 9.8 = 627.2 \]
- Now substitute back:
\[ r = \sqrt{\frac{1}{1.112 \times 10^{-10} \cdot 627.2}} \approx \sqrt{1.58 \times 10^{8}} \approx 3.97 \times 10^{4} \, \text{m} \]

### Final Answer:
The distance \( r \) should be approximately \( 39700 \, \text{m} \) or \( 39.7 \, \text{km} \).