To find the initial charges on the two identical conducting spheres, we can use Coulomb's law, which describes the electrostatic force between two charged objects. The law states that the force \( F \) between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by the formula:
Coulomb's Law
The formula is expressed as:
F = k * |q1 * q2| / r²
where:
- F is the electrostatic force between the charges (in Newtons),
- k is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \),
- q1 and q2 are the magnitudes of the charges (in Coulombs),
- r is the distance between the centers of the two charges (in meters).
Step 1: Initial Charges Calculation
Initially, the spheres attract each other with a force of \( 0.108 \, \text{N} \) when separated by \( 0.5 \, \text{m} \). We can set up the equation using Coulomb's law:
0.108 = (8.99 \times 10^9) * |q1 * q2| / (0.5)²
Rearranging this gives us:
|q1 * q2| = (0.108 * (0.5)²) / (8.99 \times 10^9)
Calculating this:
|q1 * q2| = (0.108 * 0.25) / (8.99 \times 10^9) = 2.99 \times 10^{-12} \, \text{C}²
Step 2: Charges After Connecting the Wire
When the wire connects the spheres, they share their charges equally because they are identical. Let’s denote the initial charges as \( q_1 \) and \( q_2 \). After connecting the wire, the total charge becomes \( q_1 + q_2 \), and since they are identical, each sphere will have a charge of:
q' = (q1 + q2) / 2
After the wire is removed, the spheres repel each other with a force of \( 0.0360 \, \text{N} \). Using Coulomb's law again:
0.0360 = (8.99 \times 10^9) * |q' * q'| / (0.5)²
Rearranging gives:
|q'|² = (0.0360 * (0.5)²) / (8.99 \times 10^9)
Calculating this yields:
|q'|² = (0.0360 * 0.25) / (8.99 \times 10^9) = 1.00 \times 10^{-12} \, \text{C}²
Taking the square root gives:
|q'| = 1.00 \times 10^{-6} \, \text{C}
Step 3: Finding Initial Charges
Now we have two equations:
- |q1 * q2| = 2.99 \times 10^{-12} \, \text{C}²
- |q'| = 1.00 \times 10^{-6} \, \text{C}
Substituting \( q' \) into the first equation, we have:
|q1 * q2| = |(q' + q')| = |(2 * q')| = 2 * (1.00 \times 10^{-6})² = 2.00 \times 10^{-12} \, \text{C}²
Now we can solve for \( q1 \) and \( q2 \). Let’s assume \( q1 = q \) and \( q2 = -q \) (since they attract initially). Thus:
q² = 2.99 \times 10^{-12} \Rightarrow q = \sqrt{2.99 \times 10^{-12}} \approx 1.73 \times 10^{-6} \, \text{C}
Final Results
The initial charges on the spheres were approximately:
- q1 ≈ 1.73 μC
- q2 ≈ -1.73 μC
This means one sphere had a positive charge, while the other had an equal magnitude of negative charge, leading to their initial attraction and subsequent repulsion after connecting the wire. This scenario beautifully illustrates the principles of electrostatics and charge distribution in conductors.
