Question

A charge of $+2q$ is placed at the center of an uncharged conducting shell. What will be the charges on the inner and outer surfaces of the shell, respectively? a) $-2q,+2q$ b) $-q,+q$ c) $-2q,-2q$ d) $-2q,+4q$

Short answer

Answer: The charges on the inner and outer surfaces of the shell are -2q and +2q respectively.

Step-by-step solution

Determine the initial charges of the shell

Initially, the shell is uncharged, so the charge on the inner, as well as the outer surface, is 0.

Use the principle of electrostatic induction

Now, as the charge of +2q is placed at the center of an uncharged conducting shell, it will induce an equal and opposite charge of -2q on the inner surface of the shell to neutralize the effect of the central charge. Thus, the charge on the inner surface is -2q.

Determine the outer surface charge

Since the total charge on the conducting shell remains constant, and we have induced charges of -2q on the inner surface, the outer surface will now have a charge that will maintain the initial uncharged state. So, the charges on the outer surface should be +2q.

Find the charges on the inner and outer surfaces

Therefore, the charges on the inner and outer surfaces of the shell will be $-2q$ and $+2q$ respectively. So, the correct answer is (a) $-2q,+2q$.

Key concepts

Conducting Shell

Imagine a hollow sphere made of a conductive material - this is what we call a conducting shell. It's like your everyday hollow metal ball, but in the realm of physics, it has some pretty interesting properties when it comes to electric charge.

When an electric charge is placed inside or near such a shell, it influences the free electrons in the conductor. These electrons start to move around, trying to get as far away from the charge as possible if it's the same polarity, or toward it if opposite. What's really fascinating is that the electrons redistribute themselves in such a way that they perfectly cancel out the electric field within the material of the shell itself. This means, inside the shell, it's as if the charge doesn't even exist - nullifying the field entirely. Moreover, this redistribution causes induced charges to appear on the inner and outer surfaces of the shell. Our textbook exercise demonstrates this phenomenon by showing how placing a positive charge inside the shell induces an equal but opposite charge on its inner surface.

So, as part of the exercise improvement advice, further explanation about how the conducting shell influences electron distribution is crucial. It's like watching dancers on a stage moving precisely in response to one another - each electron finds its place to keep the overall balance.

Electric Charge

The concept of electric charge is a cornerstone of physics, especially when we discuss electricity and magnetism. It’s basically a property of particles that causes them to experience a force when placed in an electric and magnetic field. Think of it like two people with strong personalities—similar types repel, opposites attract.

Charges come in two flavors: positive and negative. Protons have a positive charge, electrons have a negative charge, and neutrons are neutral. When we get into the textbook example, a positive charge of +2q is used. This 'q' represents a quantity of charge, which could be any unit, like the electron's charge or a coulomb. By introducing this charge to the center of a conducting shell, induced charges are created, and through the beauty of physics, the shell itself remains electrically neutral overall. This entire dance of attraction and repulsion is the fundamental reason your electronic devices work!

Understanding the behavior of electric charges, especially within conductors, allows students to get a better grasp of electrostatics and how forces between charges are managed and manipulated in different scenarios.

Principle of Superposition

The principle of superposition might sound like a grand concept, but it’s pretty straightforward—it’s the idea that multiple influences can peacefully coexist. Specifically, when it comes to electric charges, it tells us that the total electric field created by multiple charges is merely the vector sum of the individual fields that each charge would create if it were alone.

This principle is like a team working on a project. Each member's contribution is essential, and the final outcome is a blend of all their efforts. In our shell-and-charge scenario from the exercise, it explains how the electric fields from the inner induced charge and the central charge combine to impact the final distribution of charge on the shell's surfaces. The superposition principle reassures us that we can calculate these fields one by one and then just add them up to understand the combined effect.

It's vital for students to grasp this principle because it forms the bedrock of understanding complex interactions in electrostatics. It allows you to break down seemingly complicated problems into manageable parts, where each charge can be considered independently before merging the results for the full picture.