To solve the problem of finding the final charge on each of the two identical metallic spheres after they come into contact, we can use Coulomb's Law and the principle of charge conservation. Let's break this down step by step.
Understanding the Initial Setup
We have two metallic spheres with unequal but opposite charges. When they come into contact, the charges redistribute evenly because they are identical in size and shape. After separating them again, they will each have the same charge.
Applying Coulomb's Law
Coulomb's Law states that the force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by the formula:
F = k * |q1 * q2| / r²
where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).
Charge Redistribution
Let’s denote the initial charges of the spheres as \( q_1 \) and \( q_2 \). Since they have opposite charges, we can express them as \( q_1 = +Q \) and \( q_2 = -Q' \). When they touch, the total charge \( Q_{total} \) is:
Q_{total} = q_1 + q_2 = Q - Q'
After contact, the charge on each sphere becomes:
q_{final} = Q_{total} / 2
Calculating the Final Charge
After they are separated, the force of repulsion between them is given as 0.108 N, and the distance \( r \) is 0.50 m. We can set up the equation using Coulomb's Law:
0.108 = k * |q_{final} * q_{final}| / (0.50)²
Substituting \( k \) into the equation:
0.108 = (8.99 \times 10^9) * |q_{final}^2| / 0.25
Rearranging gives:
|q_{final}^2| = (0.108 * 0.25) / (8.99 \times 10^9)
Calculating the right side:
|q_{final}^2| = (0.027) / (8.99 \times 10^9) ≈ 3.00 \times 10^{-12}
Taking the square root to find \( q_{final} \):
q_{final} ≈ \sqrt{3.00 \times 10^{-12}} ≈ 1.73 \times 10^{-6} \, \text{C}
Final Charges on Each Sphere
Thus, after they come into contact and are separated, the final charge on each sphere is approximately:
q_{final} ≈ 1.73 \, \mu C
Since one sphere initially had a positive charge and the other a negative charge, after contact, both spheres will have a charge of approximately +1.73 µC.
Summary
In summary, after the two identical metallic spheres with unequal opposite charges come into contact and are then separated, they each end up with a final charge of about +1.73 µC. This process illustrates the principles of charge conservation and the behavior of electric forces between charged objects.
