Question

A metal sphere A has charge \(Q\). Two other spheres, B and \(\mathrm{C},\) are identical to \(\mathrm{A}\) except they have zero net charge. A touches \(\mathrm{B}\), then the two spheres are separated. B touches \(C\), then those spheres are separated. Finally, C touches A and those two spheres are separated. How much charge is on each sphere?

Short answer

A has \(\frac{3Q}{8}\), B has \(\frac{Q}{4}\), C has \(\frac{3Q}{8}\).

Step-by-step solution

Understand Initial Conditions

Initially, sphere A has a charge of \(Q\), while spheres B and C are initially neutral (0 charge).

A Touches B

When sphere A contacts sphere B, the charge redistributes evenly between them since they are identical. The total charge \(Q\) is shared equally, giving each sphere \(\frac{Q}{2}\). After separation, sphere A has \(\frac{Q}{2}\) and sphere B has \(\frac{Q}{2}\).

B Touches C

Now, sphere B (with \(\frac{Q}{2}\) charge) contacts sphere C (initially neutral). The \(\frac{Q}{2}\) charge spreads equally, giving each sphere another equal distribution. So they both end up with \(\frac{Q}{4}\) after separation.

C Touches A

Sphere C now has \(\frac{Q}{4}\), and sphere A has \(\frac{Q}{2}\). When they touch, the total charge \(\frac{Q}{2} + \frac{Q}{4} = \frac{3Q}{4}\) redistributes evenly. Each receives \(\frac{3Q}{8}\). After separation, sphere A and C both have \(\frac{3Q}{8}\).

Summarize Final Charges

After all the touching and separation processes, the charges are distributed as follows: A has \(\frac{3Q}{8}\), B still retains \(\frac{Q}{4}\), and C has \(\frac{3Q}{8}\).

Key concepts

Charge Distribution

Charge distribution is a fundamental concept in electrostatics, describing how electric charge spreads across objects when they come into contact. In the exercise involving metal spheres A, B, and C, understanding charge distribution is key to determining the final charge on each sphere.

When two conducting bodies touch, the charges redistribute due to the free movement of electrons. This redistribution aims to minimize the potential energy, leading to an equal charge per unit area among identical bodies. In the exercise, when sphere A with charge \(Q\) touches sphere B, they attain equal charges of \(\frac{Q}{2}\). Upon touching sphere B with C, the charge \(\frac{Q}{2}\) gets split further into \(\frac{Q}{4}\) for each. Finally, when C touches A, their charges combine and redistribute to \(\frac{3Q}{8}\) each. This step-by-step redistribution showcases the underlying principle of charge balancing, highlighting how charges strive for equilibrium when shared among identical conductive surfaces.

Metal Spheres

Metal spheres in electrostatics serve as an excellent model for exploring charge interactions because of their conductive nature. Conductors like metal spheres allow free movement of electrons, which facilitates charge transfer and redistribution.

In the given exercise, the metal spheres A, B, and C are all conductive and identical in size and material. This identity ensures that when they touch, the charge distributes evenly due to their equal capacitance. The equal surface area of these spheres plays a critical role in making the charge distribution simple and predictable. This predictable behavior forms the basis for many experiments and helps students understand the principles of electrostatics and charge equilibrium. Such illustrations show why scientists and engineers often choose metal spheres for studying electrical charge phenomena.

Charge Conservation

Charge conservation is a pivotal principle in physics, stating that the total electric charge in an isolated system remains constant. Despite the transfer of charge between objects within the system, the overall charge does not change.

In our metal sphere problem, even as spheres A, B, and C come into contact and share charge, the total system charge remains \(Q\). Initially, sphere A holds the charge \(Q\), and B and C have none. After each touch and separation, the key observation is that the summation of all charges across the spheres stays \(Q\):

• Initially: \(Q + 0 + 0 = Q\)
• After A and B touch: \(\frac{Q}{2} + \frac{Q}{2} + 0 = Q\)
• After B and C touch: \(\frac{Q}{2} + \frac{Q}{4} + \frac{Q}{4} = Q\)
• After C and A touch: \(\frac{3Q}{8} + \frac{Q}{4} + \frac{3Q}{8} = Q\)

This exercise clearly illustrates the law of conservation of charge, demonstrating that regardless of the movements or interactions, the charges merely rearrange between the bodies without any loss or gain in the total system charge.

Electric Charge Interactions

Electric charge interactions refer to how charges on different objects act upon each other through electrostatic forces. These interactions have observable effects when objects come into close proximity or make contact.

In the scenario with spheres A, B, and C, the interaction occurs when these spheres touch each other, allowing charge to flow and redistribute. The electric charges on these conductive spheres redistribute due to repulsion between like charges and the tendency of electrons to balance uniformly over a conductive surface. When sphere A, initially charged, touches B, the electric field inside the spheres drives the charges to spread evenly, stabilizing at equal distributions. This balancing act continues in subsequent interactions between B and C, then C and A.

These interactions reveal the invisible forces acting at the atomic level, governed by fundamental electrostatic principles, showcasing that charges move to minimize repulsion and ensure a uniform potential, ultimately stabilizing the charge distribution. Such interactions are foundational in understanding phenomena ranging from simple static electricity to complex electrical circuit functionality.