Problem 21 - Fundamentals of Physics

Question

An isolated conductor has net charge \(+10 \times 10^{-6} \mathrm{C}\) and a cavity with a particle of charge \(q=+3.0 \times 10^{-6} \mathrm{C}\). What is the charge on (a) the cavity wall and (b) the outer surface?

Step-by-Step Solution

Answer

(a) Charge on the cavity wall is \(-3.0 \times 10^{-6} \mathrm{C}\); (b) charge on the outer surface is \(+13.0 \times 10^{-6} \mathrm{C}\).

Step 1: Understanding the Concept of Charge Conservation

An isolated conductor undergoes charge redistribution so that the net charge remains the same. Given the net charge of the conductor is \(+10 \times 10^{-6} \mathrm{C}\) and there is a cavity with a charge \(q=+3.0 \times 10^{-6} \mathrm{C}\) inside it. When a charge is inside a conductor completely surrounded by it, this charge induces an equal and opposite charge on the inner surface of the cavity.

Step 2: Determining the Charge on the Cavity Wall

The particle with charge \(+3.0 \times 10^{-6} \mathrm{C}\) inside the cavity induces a charge of \(-3.0 \times 10^{-6} \mathrm{C}\) on the cavity wall to maintain equilibrium and neutrality of the conductor's internal field.

Step 3: Calculating the Charge on the Outer Surface

The total charge on the entire conductor is \(+10 \times 10^{-6} \mathrm{C}\). The charge on the cavity wall is \(-3.0 \times 10^{-6} \mathrm{C}\), so by conservation of charge, the outer surface must have a charge that is the sum of these, which is \(+10 \times 10^{-6} \mathrm{C} - (-3.0 \times 10^{-6} \mathrm{C}) = +13.0 \times 10^{-6} \mathrm{C}\).

Key Concepts

Conservation of Charge

The principle of conservation of charge is fundamental in physics. It states that the total electric charge in an isolated system remains constant. This means charge can neither be created nor destroyed, only transferred from one object to another. In an isolated conductor, where no external charge can enter or leave, the total charge is preserved. For example, if a conductor has a net charge of \(+10 \times 10^{-6}\,\mathrm{C}\) and there is a cavity inside it with a charge \(q=+3.0 \times 10^{-6}\,\mathrm{C}\), the overall charge of \(+10 \times 10^{-6}\,\mathrm{C}\) is maintained. Any internal changes, such as induction of charges, do not alter this total. This principle ensures that when charges move or redistribute within the conductor, the net charge remains consistent.

Induced Charge

Induced charge is a result of electrostatic influence. When a charged object is placed inside a conductor, it causes a redistribution of charge within the conductor. Specifically, a charge within a cavity of a conductor will induce an opposite charge on the surface of the cavity wall. Consider a particle inside a conductor's cavity with a charge of \(+3.0 \times 10^{-6}\,\mathrm{C}\). To balance the internal electric field and ensure neutrality, the cavity wall will have an induced charge of \(-3.0 \times 10^{-6}\,\mathrm{C}\).
This opposite charge is not "new" charge but a rearrangement of existing charges within the conductor. The shift ensures that the space outside the cavity remains unaffected by the internal charge. Induced charges are essential for creating balanced electrical environments within a material while complying with the conservation of charge.

Conductor Properties

Conductors have unique properties that allow them to efficiently manage electrical charge. One main feature is their ability to permit free movement of electrons throughout their structure. This mobility results in a quick redistribution of charges in response to internal or external electric fields. In our scenario, the outer surface of the conductor must have a charge that complements the cavity’s induced charge and maintains the total charge conservation across the entire conductor.
Given the conductor has a net charge of \(+10 \times 10^{-6}\,\mathrm{C}\), and the cavity wall carries an induced charge of \(-3.0 \times 10^{-6}\,\mathrm{C}\), the outer surface, by necessity, holds the remaining charge: \(+13.0 \times 10^{-6}\,\mathrm{C}\). This distribution reflects the multipart ability of conductors to spread charge uniformly across their surfaces, ensuring minimal electric field within the conductor itself beyond the surface layer. This surface effect is an intrinsic property of conductors.