Problem 5 - Physics

Question

Consider three identical metal spheres, \(A, B,\) and \(C .\) Sphere A carries a charge of \(+5 q .\) Sphere \(B\) carries a charge of \(-q\). Sphere \(\mathrm{C}\) carries no net charge. Spheres \(\mathrm{A}\) and \(\mathrm{B}\) are touched together and then separated. Sphere \(\mathrm{C}\) is then touched to sphere \(A\) and separated from it. Last, sphere \(C\) is touched to sphere \(B\) and separated from it. (a) How much charge ends up on sphere \(\mathrm{C}\) ? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

Step-by-Step Solution

Answer

C ends up with \(1.5q\). Total charge initially: \(4q\). Total charge after: \(4q\).

Step 1: Initial Charges on Spheres

Initially, the charge on sphere A is \(+5q\), on sphere B is \(-q\), and sphere C has no charge \(0\).

Step 2: Calculate Total Initial Charge

The total initial charge is the sum of charges on all three spheres:\[+5q + (-q) + 0 = +4q.\]

Step 3: Spheres A and B Touch Each Other

When two spheres touch, they share their charges equally because they are identical. After A and B touch, the total charge is \(+5q - q = +4q\). This charge is equally divided between them:\[\text{Charge on A} = \text{Charge on B} = \frac{4q}{2} = 2q.\]

Step 4: Sphere C Touches Sphere A

Sphere C, initially with 0 charge, touches sphere A which now has a charge of \(+2q\). The total charge shared between them is \(2q + 0 = 2q\), so each will have \(\frac{2q}{2} = q\) after separating.

Step 5: Sphere C Touches Sphere B

Sphere C now has a charge of \(q\) and touches sphere B (also \(2q\)). The total charge is \(q + 2q = 3q\). This divides equally, leaving each sphere with \[\frac{3q}{2} = 1.5q.\]

Key Concepts

Charge distribution

Charge distribution is a fundamental concept in electrostatics, which explains how electric charge is spread over objects. In the context of the exercise, the three identical metal spheres, A, B, and C, exhibit different initial charges. Sphere A starts with a charge of \(+5q\), sphere B with \(-q\), and sphere C is initially neutral with no charge. When two charged conductors touch, their charges redistribute until equilibrium is reached. This means that the total charge will be equally shared between the spheres if they are identical. For instance, when sphere A and B, having combined charges of \(+4q\), touch, each ends with a charge of \(2q\). The physical touch causes electrons to move between the objects to balance forces, ensuring that each sphere has the same charge due to their identical nature and properties. This charge sharing reflects the underlying principle that charges redistribute to minimize potential energy across conductive materials, which is a direct consequence of electrostatic forces and charge uniformity.

Conservation of charge

The law of conservation of charge is crucial in all electrostatic processes. It states that the total charge in an isolated system remains constant. Therefore, regardless of how charges are distributed or transferred between objects, the total charge doesn't change. In this exercise, the initial total charge of the three spheres is \(+4q\), calculated as:

• Sphere A: \(+5q\)
• Sphere B: \(-q\)
• Sphere C: \(0\)

This total of \(+4q\) stays constant throughout the interactions, even as sphere C goes through a series of touches and separations with spheres A and B. After each interaction, such as when spheres A and B touch, they redistribute their charge but do not lose or gain any charge externally. Similarly, after touching sphere C to spheres A and B, all transformations respect the conservation rule. Ultimately, the total charge of the system remains \(+4q\) after all interactions, demonstrating this fundamental electrostatic principle. Knowing and applying the conservation of charge ensures accurate predictions of final charge distributions on objects.

Identical conductors

Identical conductors play a vital role in understanding charge distribution and interaction. Because the spheres A, B, and C are identical, they have the same physical properties such as size, shape, and capacity to hold charge. Identical properties ensure that when any two spheres touch, charge is evenly distributed between them.This equal distribution of charge is due to the identical ability of each conductor to hold charge when they share a common point. This simplifies calculations; when two identical conductors touch, they share their total charge equally. For example, after spheres A and B touch and redistribute their total of \(+4q\), each ends up with \(2q\). When sphere C, initially neutral, touches sphere A (which now holds \(2q\)), the charges will again equalize, and both end up with \(q\).Understanding the identical nature of these conductors helps predict that subsequent touches will always lead to an equal split of total charge at that interaction's moment. This property heavily simplifies analyzing interactions among conductive objects, especially in electrostatics, where different shape or size would otherwise complicate charge calculations.