Identical isolated conducting spheres 1 and 2 have equal charges and are separated by a distance that is large compared with their diameters. The electrostatic force acting on sphere 2 due to sphere 1 is F. Suppose now that a third identical sphere 3, having an insulating handle and initially neutral, is touched first to sphere 1, then to sphere 2, and finally removed. The electrostatic force that now acts on sphere 2 has magnitude F'. What is the ratio F'/F?
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To find the ratio F'/F, we need to analyze the charge distribution on the spheres after sphere 3 is touched to sphere 1 and sphere 2.
Initially, both sphere 1 and sphere 2 have equal charges (let's call it Q) and are separated by a large distance. The electrostatic force between them can be calculated using Coulomb's law:
F = k(Q^2 / r^2)
where:
F is the electrostatic force,
k is the electrostatic constant (9 x 10^9 Nm^2/C^2),
Q is the charge on each sphere, and
r is the distance between the spheres.
When sphere 3 is touched to sphere 1, it acquires some charge (let's call it q) through the process of charge redistribution. Since spheres 1 and 3 are identical and in contact, they will end up with equal charge q/2 on each sphere.
Now, when sphere 3 is touched to sphere 2, the charge q/2 on sphere 3 will redistribute again. However, this time the charge on sphere 2 will repel the charge on sphere 3 since they have the same charge polarity. As a result, sphere 2 will gain a negative charge (-q/2) and sphere 3 will acquire a positive charge (+q/2). Sphere 1 remains unchanged.
After this process, we have:
- Sphere 1: charge Q
- Sphere 2: charge -q/2
- Sphere 3: charge +q/2
The electrostatic force (F') between sphere 1 and sphere 2 can now be calculated using Coulomb's law with their new charges:
F' = k(Q * (-q/2) / r^2)
To find the ratio F'/F, we can substitute the expressions for F and F' and cancel out the common terms:
F'/F = (Q * (-q/2))/(Q^2) = -q/(2Q)
Therefore, the ratio of F'/F is -q/(2Q).
To determine the ratio F'/F, we need to consider the effect of touching sphere 3 to sphere 1 and sphere 2.
Initially, spheres 1 and 2 are identical and have equal charges. Let's assume they have charge Q each. Since the spheres are isolated and have charges of the same magnitude, they will repel each other with equal force F due to electrostatic interaction.
When sphere 3 is touched to sphere 1, some of the charge from sphere 1 will transfer to sphere 3. The charges on sphere 1 and sphere 2 will now be reduced.
Next, when sphere 3, which now carries charge from sphere 1, is touched to sphere 2, some of the charge will be transferred from sphere 3 to sphere 2. Now, sphere 2 will have more charge than sphere 1.
Finally, when the insulating handle of sphere 3 is removed, spheres 1, 2, and 3 are isolated with their new charge distributions.
The electrostatic force between spheres 1 and 2 can be calculated using Coulomb's law. The force between two point charges is given by:
F = (k * |q1 * q2|) / r^2,
where F is the force, k is the electrostatic constant, q1 and q2 are the charges on the two spheres, and r is the separation between them.
Let's assume the new charge on sphere 1 is q1' and the new charge on sphere 2 is q2'. Since we don't know the exact charge distributions, we will consider the ratio of their charges.
The ratio F'/F will be equal to:
F' / F = ((k * |q1' * q2'|) / r^2) / ((k * |q1 * q2|) / r^2)
= (|q1' * q2'|) / (|q1 * q2|)
= (q1' * q2') / (q1 * q2)
Now, since the spheres are identical and sphere 3 is initially neutral, the charges transferred from sphere 1 to sphere 3 and from sphere 3 to sphere 2 are equal in magnitude. Let's denote this charge as ΔQ.
Thus, q1' = q1 - ΔQ and q2' = q2 + ΔQ.
Substituting these values into the ratio expression:
F' / F = ((q1 - ΔQ) * (q2 + ΔQ)) / (q1 * q2)
= (q1q2 + q1ΔQ - q2ΔQ - ΔQ^2) / (q1 * q2)
= 1 + ((ΔQ - ΔQ^2) / (q1 * q2))
Now, if we assume that ΔQ is small compared to q1 and q2, we can ignore the ΔQ^2 term. This assumption is valid for a large separation distance compared to sphere diameters.
Therefore, the ratio F'/F simplifies to:
F' / F ≈ 1 + ΔQ / (q1 * q2)
Since ΔQ is the charge transferred, we can consider it as a fraction of the original charge, Q:
ΔQ = kQ,
where k is a fraction between 0 and 1.
Substituting this value:
F' / F ≈ 1 + kQ / (q1 * q2)
Now, since spheres 1 and 2 initially have the same charge, q1 = q2 = Q. Substituting these values:
F' / F ≈ 1 + kQ / (Q * Q)
= 1 + k / Q
Hence, the final ratio F'/F is approximately equal to 1 + k / Q.
Note that without knowing the exact values of ΔQ, k, and Q, we can't evaluate this ratio numerically. However, we can conclude that the ratio is slightly greater than 1.