There are two identical, positively charged conducting spheres fixed in space. The spheres are 44.6 cm apart (center to center) and repel each other with an electrostatic force of F1 = 0.0675 N. Then, a thin conducting wire connects the spheres, redistributing the charge on each sphere. When the wire is removed the spheres still repel but with a force of F2 = 0.115 N. Using this information, find the initial charge on each sphere, q1 and q2 if initially q1 < q2.
To find the initial charge on each sphere, q1 and q2, we can make use of Coulomb's law, which states that the electrostatic force between two charged objects is given by the equation:
F = (k * |q1 * q2|) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (9 × 10^9 N m^2/C^2)
|q1| and |q2| are the magnitudes of the charges on the two spheres
r is the distance between the centers of the two spheres
Given that the spheres are identical and initially repel each other with a force of F1 = 0.0675 N when they are 44.6 cm apart, we can write an equation for this situation:
0.0675 = (9 × 10^9 * |q1 * q2|) / (0.446)^2 -- Equation 1
After the thin conducting wire connects the spheres and redistributes the charge on each sphere, the charges on the spheres will be equal since they have been connected, let's call this new charge q_final. When the wire is removed, the spheres repel each other with a force of F2 = 0.115 N. We can write an equation for this situation too:
0.115 = (9 × 10^9 * |q_final * q_final|) / (0.446)^2 -- Equation 2
Since q_final is the same for both spheres, we can rewrite Equation 2 as follows:
0.115 = (9 × 10^9 * (q_final)^2) / (0.446)^2
Simplifying this equation, we find:
(q_final)^2 = (0.115 * (0.446)^2) / (9 × 10^9)
(q_final)^2 = 0.00000027973
Taking the square root of both sides gives:
q_final = ±0.000529 C (Note that the magnitude of q_final is taken, as mentioned earlier)
Since initially, q1 < q2, we have the following relationship:
q1 + q_final = q2 - q_final
Substituting the values of q_final, we get:
q1 + 0.000529 = q2 - 0.000529
Simplifying this equation, we find:
q2 = q1 + 0.001058
Now, substituting the value of q2 in Equation 1, we can solve for q1:
0.0675 = (9 × 10^9 * |q1 * (q1 + 0.001058)|) / (0.446)^2
Simplifying this equation, we find:
0.0675 = (9 × 10^9 * (q1 * (q1 + 0.001058))) / (0.446)^2
Solving this equation will give us the value of q1.