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Consider three identical metal spheres, A, B, and 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 \(A\) and \(B\) are touched together and then separated. Sphere C is then touched to sphere A and separated from it. Last, sphere \(C\) is touched to sphere \(\mathrm{B}\) and separated from it. (a) How much charge ends up on sphere 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?

Short Answer

Expert verified
Sphere C ends with \(\frac{3}{2}q\) charge. Total charge is \(+4q\) before and after touching.

Step by step solution

01

Calculate Initial Total Charge

First, determine the total initial charge of the three spheres. Sphere A has a charge of \(+5q\), sphere B has a charge of \(-q\), and sphere C has \(0\) charge. Thus, the total is:\[+5q - q + 0 = +4q\]
02

Sphere A and Sphere B Touch and Separate

When spheres A and B are touched, they will share their total charge equally. Initially, the total charge of \(+5q\) and \(-q\) amounts to \(+4q\). Since they are identical and share the charge equally, each sphere ends up with:\[\frac{+4q}{2} = +2q\]Thus, after separation, Sphere A and Sphere B both have \(+2q\).
03

Sphere A and Sphere C Touch and Separate

Next, Sphere C (initially \(0\) charge) touches Sphere A (\(+2q\)) and they share their total charge equally. Combined, they have \(+2q\). After touching, each sphere has:\[\frac{+2q}{2} = +q\]Thus, Sphere A and Sphere C each have \(+q\) after separation.
04

Sphere C and Sphere B Touch and Separate

Finally, Sphere C (\(+q\)) touches Sphere B (\(+2q\)), and they share their total charge equally. The total charge is \(+3q\) which they share equally, giving:\[\frac{+3q}{2} = \frac{3}{2}q\]Therefore, Sphere C and Sphere B each have \(\frac{3}{2}q\) after separation.
05

Total Charge Before and After

Before the spheres are touched, the total charge across all three spheres was \(+4q\). After all touching and separations, the total charge remains the same. - Before touching, charges are: A (\(+5q\)), B (\(-q\)), C (\(0\)).- After touching, charges are: A (\(+q\)), B (\(\frac{3}{2}q\)), C (\(\frac{3}{2}q\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Charge Distribution
In electrostatics, charge distribution refers to how charge is spread across different objects when they interact. When objects like the metal spheres in our exercise are touched, they redistribute their total charge among themselves. This process usually happens because the objects strive for an equilibrium, meaning that they attempt to have equal charges if they are identical.
In the initial step, when Sphere A and Sphere B touch, their charges combine and then distribute evenly between them. Since they are identical, they end up with the same charge after separation. This ability to share and redistribute charge is fundamental to understanding electrostatic interactions among conductive objects. The charge distribution process explained here assumes a uniform conductive surface, leading to an even spread of charges.
Conservation of Charge
The principle of conservation of charge is a cornerstone in electrostatics. It states that the total charge in an isolated system remains constant, despite any redistribution of charge within the system. In the context of the problem, the total initial charge among the three spheres is calculated as \[+5q - q + 0 = +4q.\]
No matter how the spheres are touched or moved, the total charge before and after touching remains equal: \[+4q.\]
This principle holds true universally, meaning that within a closed system, charge can be neither created nor destroyed. It can only move from one object to another. In practical terms, this means that while individual objects may change in their charge state due to interactions, the worldwide total charge remains unchanged. This law helps ensure predictable outcomes whenever we analyze electrostatic scenarios.
Conductors
Conductors are materials that easily allow the flow of electric charge. Metals, like the spheres in our exercise, are good conductors due to their electrons being loosely bound and capable of moving freely. This characteristic enables conductors to distribute charge across their surfaces quickly upon contact with other objects.
In the exercise, when any two spheres touch, the electrons freely flow between them until they each carry an equal charge. This behavior is crucial for the accurate analysis of electrostatic interactions in conductive materials. Conductors are essential in understanding how charge equilibrium is achieved, demonstrating the ability of charge to be restocked between objects readily. Understanding conductors' role in charge distribution and conservation is vital, as it provides insights into how charges equalize in conductive systems.

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Most popular questions from this chapter

You and your team are designing a device that can be used to position a small, plastic object in the region between the plates of a parallel-plate capacitor. A small plastic sphere of mass \(m=1.20 \times 10^{-2} \mathrm{kg}\) carries a charge \(q=+0.200 \mu \mathrm{C}\) and hangs vertically (along the \(y\) direction) from a massless, insulating thread (length \(l=10.0 \mathrm{cm})\) between two vertical capacitor plates. When there is no electric field, the object resides at the midpoint between the plates (at \(x=0\) ). However, when there is a field between plates (in the \(\pm x\) direction) the object moves to a new equilibrium position. (a) To what value should you set the field if you want the object to be located at \(x=2.10 \mathrm{cm} ?\) (b) To what value should you set the field if you want the object to be located at \(x=-3.30 \mathrm{cm} ?\)

A surface completely surrounds a \(+2.0 \times 10^{-6} \mathrm{C}\) charge. Find the electric flux through this surface when the surface is (a) a sphere with a radius of \(0.50 \mathrm{m},\) (b) a sphere with a radius of \(0.25 \mathrm{m},\) and \((\mathrm{c})\) a cube with edges that are \(0.25 \mathrm{m}\) long.

The total electric field \(\overrightarrow{\mathbf{E}}\) consists of the vector sum of two parts. One part has a magnitude of \(E_{1}=1200 \mathrm{N} / \mathrm{C}\) and points at an angle \(\theta_{1}=35^{\circ}\) above the \(+x\) axis. The other part has a magnitude of \(E_{2}=1700\) \(\mathrm{N} / \mathrm{C}\) and points at an angle \(\theta_{2}=55^{\circ}\) above the \(+x\) axis. Find the magnitude and direction of the total field. Specify the directional angle relative to the \(+x\) axis.

An electrically neutral model airplane is flying in a horizontal circle on a \(3.0-\mathrm{m}\) guideline, which is nearly parallel to the ground. The line breaks when the kinetic energy of the plane is 50.0 J. Reconsider the same situation, except that now there is a point charge of \(+q\) on the plane and a point charge of \(-q\) at the other end of the guideline. In this case, the line breaks when the kinetic energy of the plane is 51.8 J. Find the magnitude of the charges.

Two very small spheres are initially neutral and separated by a distance of \(0.50 \mathrm{m} .\) Suppose that \(3.0 \times 10^{13}\) electrons are removed from one sphere and placed on the other. (a) What is the magnitude of the electrostatic force that acts on each sphere? (b) Is the force attractive or repulsive? Why?

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