To solve this problem, we need to analyze the interactions between the charged and uncharged spheres step by step. The scenario involves three identical metallic spheres, A, B, and C, and a fourth charged sphere that influences the system. Let's break it down systematically.
Initial Setup
We start with three uncharged metallic spheres A, B, and C, positioned at the corners of an equilateral triangle with a side length much larger than their radius (d >> a). When the charged sphere touches sphere A, it transfers some of its charge due to electrostatic induction.
Charge Transfer to Sphere A
When the charged sphere with charge q touches sphere A, charge will redistribute between them. Since A and the fourth sphere are identical, they will share the charge equally. Therefore, after the touch:
- Charge on A = q/2
- Charge on the fourth sphere = q/2
Grounding Sphere B
Next, sphere B is earthed. Grounding allows charge to flow between the sphere and the Earth, which has a vast reservoir of charge. Since sphere B is initially uncharged, it will attract some of the positive charge from sphere A due to the induced charge on A.
Induction Effect
Sphere A, now having a charge of q/2, will induce a negative charge on sphere B when it is earthed. The amount of charge that flows to B can be calculated using the principle of electrostatic induction:
- Charge on B after grounding = -q/4 (since it will attract half of the positive charge from A)
- Charge on A after B is earthed = q/2 + q/4 = 3q/4
Removing the Ground Connection from B
Once the grounding connection is removed, sphere B retains its charge of -q/4. It is now isolated, and its charge will not change unless influenced by another charged object.
Grounding Sphere C
Now, sphere C is earthed. Initially, it is uncharged, but it will also be influenced by the charge on sphere A (which is now 3q/4). The presence of this charge will induce a charge on sphere C:
- Charge on C while earthed = +q/4 (it will attract the positive charge from A)
- Charge on A after C is earthed = 3q/4 - q/4 = q/2
Final Charge on Sphere C
After grounding C, it will have a charge of +q/4. When the grounding is removed, sphere C retains this charge. Therefore, the final charge on sphere C is:
In summary, through a series of interactions involving charge transfer and grounding, we find that sphere C ends up with a charge of +q/4 after the entire process is completed. This illustrates the principles of electrostatics, including charge induction and the effects of grounding on metallic conductors.
