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Chapter 17: Problem 61

A neutral conductor completely encloses a hole inside of it. You observe that the outer surface of this conductor carries a charge of \(-12 \mu \mathrm{C}\). (a) Can you conclude that there is a charge inside the hole? If so, what is this charge? (b) How much charge is on the inner surface of the conductor?

Short Answer

Expert verified
(a) Yes, the charge in the hole is +12 碌C. (b) The charge on the inner surface is -12 碌C.

Step by step solution

01

Understanding the Problem

We have a neutral conductor with a hole and an outer surface charge of \(-12 \, \mu \text{C}\). We need to determine if there is a charge inside the hole and calculate the charge on the inner surface of the conductor.
02

Applying Gauss's Law

According to Gauss's Law, the total charge enclosed by a Gaussian surface that lies just inside the conductor's surface should equal the charge on the conductor's surface. Since the conductor as a whole is neutral and there's an outer surface charge of \(-12 \, \mu \text{C}\), there must be a corresponding positive charge inside the hole.
03

Finding the Charge Inside the Hole

To maintain the neutrality of the conductor as a whole, there must be a charge of \(+12 \, \mu \text{C}\) inside the hole to balance the \(-12 \, \mu \text{C}\) on the outer surface.
04

Calculating the Charge on the Inner Surface

If there is a charge of \(+12 \, \mu \text{C}\) inside the hole, it induces an equal and opposite charge on the inner surface of the conductor to maintain neutrality within the conductor itself. Thus, the charge on the inner surface must be \(-12 \, \mu \text{C}\).

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Key Concepts

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

Conductors and Charge Distribution
Conductors are materials that allow the free movement of electric charge, usually electrons, within them. This property makes conductors excellent at redistributing charge, which is key in understanding many electrical phenomena. When a conductor is subjected to an external electric field, free charges move until they reach an equilibrium distribution. This means the electric field within the conductor becomes zero because any internal field would cause further movement of charge.
  • Charge on the surface: Charges reside on the surface to minimize repulsive forces.
  • Equilibrium: Charge distribution achieves equilibrium with zero internal electric field.
Understanding how charges distribute themselves on conductors helps predict and control electric interactions in various applications.
Electrostatics
Electrostatics involves the study of electric charges at rest. Unlike current electricity, where charges are in motion, electrostatic phenomena occur when charges are stationary, leading to interactions governed by Coulomb's Law and electric fields.
  • Static Charges: These do not flow like a current but exert forces on each other.
  • Electric Field: A vector field around a charged particle influencing other charges.
Gauss's Law is a pivotal principle in electrostatics. It relates the electric flux through a closed surface to the charge enclosed by that surface. This is particularly helpful when analyzing conductors since it simplifies the process of calculating charge distributions.
Neutral Conductors
A neutral conductor has no net charge, meaning that the total positive charge equals the total negative charge. However, it's essential to understand that even though the conductor is neutral overall, charges can redistribute themselves, especially if there's an influence from external charges or fields.
  • Charge Redistribution: Charges move to achieve stability, even in neutral conductors.
  • Influence of External Charges: External fields can induce charge separation within the conductor.
In the given exercise, a neutral conductor ends up with an outer surface charge of \(-12 \mu \text{C}\) which indicates a form of charge separation or induction, while the net charge remains zero.
Charge Balance in Conductors
The concept of charge balance in conductors hinges on maintaining neutrality or balancing internal and external charges. If a conductor starts neutral, the sum of all charges within and on its surface must remain zero.
  • Outer Surface Charge: Any imbalance on the outer surface suggests an internal charge needs to balance it.
  • Induced Charge: Inner surfaces can host charges opposite to those inside cavities or external influences.
In the discussed problem, the fact that the outer surface has a charge of \(-12 \mu \text{C}\) implies there's a hidden charge within the hole. To reconcile the conductor's neutrality, a positive charge of \(+12 \mu \text{C}\) must exist inside the hole, while \(-12 \mu \text{C}\) appears on the inner surface, maintaining the charge balance.

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Most popular questions from this chapter

In a follow-up experiment, a charge of \(+40 \mathrm{pC}\) was placed at the center of an artificial flower at the end of a \(30-\mathrm{cm}\) -long stem. Bees were observed to approach no closer than \(15 \mathrm{~cm}\) from the center of this flower before they flew away. This observation suggests that the smallest external electric field to which bees may be sensitive is closest to which of these values? A. \(2.4 \mathrm{~N} / \mathrm{C}\) B. \(16 \mathrm{~N} / \mathrm{C}\) C. \(2.7 \times 10^{-10} \mathrm{~N} / \mathrm{C}\) D. \(4.8 \times 10^{-10} \mathrm{~N} / \mathrm{C}\)

A charge of \(-3.00 \mathrm{nC}\) is placed at the origin of an \(x-y\) coordinate system, and a charge of \(2.00 \mathrm{nC}\) is placed on the \(y\) axis at \(y=4.00 \mathrm{~cm} .\) (a) If a third charge, of \(5.00 \mathrm{nC},\) is now placed at the point \(x=3.00 \mathrm{~cm}, y=4.00 \mathrm{~cm},\) find the \(x\) and \(y\) components of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

\(\mathrm{A}-5.00 \mathrm{nC}\) point charge is on the \(x\) axis at \(x=1.20 \mathrm{~m}\). A second point charge \(Q\) is on the \(x\) axis at \(-0.600 \mathrm{~m}\). What must be the sign and magnitude of \(Q\) for the resultant electric field at the origin to be (a) \(45.0 \mathrm{~N} / \mathrm{C}\) in the \(+x\) direction, (b) \(45.0 \mathrm{~N} / \mathrm{C}\) in the \(-x\) direction?

Neurons are components of the nervous system of the body that transmit signals as electrical impulses travel along their length. These impulses propagate when charge suddenly rushes into and then out of a part of the neuron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately \(5.6 \times 10^{11} \mathrm{Na}^{+}\) (sodium ions) per meter, each with charge \(+e,\) enter the axon. How many coulombs of charge enter a \(1.5 \mathrm{~cm}\) length of the axon during this process?

Which of the following is true about \(\vec{E}\) inside a negatively charged sphere as described here? A. It points from the center of the sphere to the surface and is largest at the center. B. It points from the surface to the center of the sphere and is largest at the surface. C. It is zero. D. It is constant but is not zero.
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