Problem 5 - Physics

Question

Consider three identical metal spheres, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Sphere A carries a charge of \(+5 q .\) Sphere \(\mathrm{B}\) 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

(a) Charge on C ends at \(+\frac{3q}{2}\). (b) Total charge before: \(4q\). (c) Total charge after: \(4q\).

Step 1: Calculate Initial Total Charge

First, sum up the initial charges on all three spheres. Sphere A initially has a charge of \(+5q\), sphere B has \(-q\), and sphere C has no charge. Thus, the total initial charge is:\[ +5q + (-q) + 0 = 4q \]

Step 2: Spheres A and B Touch

When spheres A and B are touched together, charge redistributes between them, meaning they have to share the total charge equally because they are identical. Initially, the total charge they have is \(+5q + (-q) = 4q\). Since they are identical, each sphere will have half of the total charge:\[ \frac{4q}{2} = +2q \] So, after separation, sphere A has \(+2q\) and sphere B also has \(+2q\).

Step 3: Sphere C Touches A

Now, we touch sphere C (0 charge) to sphere A (+2q). The total charge they share is \(+2q\) and distributing it equally:\[ \frac{2q}{2} = +q \] After separation, sphere A has \(+q\) and sphere C has \(+q\).

Step 4: Sphere C Touches B

Lastly, sphere C (+q) touches sphere B (+2q). The total charge they share is \(+q + 2q = 3q\). Distributing it equally:\[ \frac{3q}{2} = +\frac{3q}{2} \] After separation, sphere B has \(+\frac{3q}{2}\) and sphere C also has \(+\frac{3q}{2}\).

Step 5: Verify Total Charge After All Actions

Sum the final charges on all spheres. Sphere A has \(+q\), sphere B has \(+\frac{3q}{2}\), and sphere C has \(+\frac{3q}{2}\). Adding these:\[ +q + \frac{3q}{2} + \frac{3q}{2} = q + 3q = 4q \] The total charge remains \(4q\).

Key Concepts

Metal Spheres

Metal spheres are intriguing objects in physics, especially when exploring the behavior of electric charges. They are often used in experiments because they have unique properties that allow charges to move freely across their surfaces. These spheres are conductive, meaning they let electrons easily pass through and redistribute. This conductance plays a significant role when they come into contact with each other.
Their ability to facilitate charge movement makes them ideal for studying various phenomena in electrostatics. The process of these charges redistributing upon contact showcases how conductors behave differently from insulators, providing valuable insight into the world of electricity. Understanding how metal spheres interact allows learners to grasp fundamental concepts that apply to countless other areas in physics.

Electric Charge

Electric charge, a basic property of matter, is central to understanding how objects interact electrically. There are two types of charges—positive and negative. Opposite charges attract each other, while like charges repel. This attraction and repulsion principle is fundamental to electric interactions.
In the case of our metal spheres, each sphere starts with a specified amount of charge. Sphere A holds a charge of "+5q," while Sphere B carries "-q." Sphere C begins as neutral, with no charge. The total combined charge, before any of them touch, is known to be the sum of these individual charges. This total helps when redistributing charges, ensuring the conservation of charge rule is respected, meaning the initial total charge remains constant through all interactions.

Charge Redistribution

Charge redistribution is a crucial concept, where charges rearrange themselves upon contact between conductive materials. When two metal spheres are brought into contact, they share their charges due to their conductive nature. They aim to equalize the charge distribution wherever possible because of their identical size and material properties.
For example, as described in the exercise, when spheres A and B are touched together, they share a combined charge of "+4q." Because they are identical, they split this charge equally, each ending up with "+2q." This same process follows through when sphere C interacts with spheres A and B. This constant redistribution is an excellent demonstration of charge conservation and equilibrium seeking in physics.

Identical Spheres

The identical nature of spheres in this exercise refers to their equal radii and identical material composition. This uniformity matters because it guarantees that any charge they acquire or share must be equitably distributed when they are in contact.
If the spheres were not identical, the distribution of charge would differ because larger or differently composed spheres might hold a different amount of charge. But in this scenario, because the spheres are identical in every way, they allow for straightforward calculations by dividing the total charge between them equally whenever they touch. This simplifies problems like this in physics, making it easier to predict and understand outcomes.