Problem 62 - Fundamentals of Physics

Question

A particle of charge \(q=1.0 \times 10^{-7} \mathrm{C}\) is at the center of a spherical cavity of radius \(3.0 \mathrm{~cm}\) in a chunk of metal. Find the electric field (a) \(1.5 \mathrm{~cm}\) from the cavity center and (b) anyplace in the metal.

Step-by-Step Solution

Answer

(a) 3.994 x 10⁵ N/C, (b) zero.

Step 1: Understanding the setup

The problem involves a charged particle located at the center of a spherical cavity within a metal. The charge is given as \( q = 1.0 \times 10^{-7} \mathrm{C} \). This charge will generate an electric field.

Step 2: Conceptualizing inside a conductor

A metal or conductor rearranges its free electrons to negate the internal electric field resulting from any charge. Therefore, inside the conductor (outside the cavity's inner surface), the electric field is zero due to electrostatic shielding. Particularly for point (b), anywhere in the metal, the electric field will be zero.

Step 3: Calculating electric field inside the cavity

The electric field at a point inside the cavity, at a distance from the charge, can be determined using Coulomb’s law. This is valid because there are no additional charges between the point and the source charge at the cavity's center.

Step 4: Applying Coulomb's Law for (a)

Use Coulomb's Law to find the electric field at a point 1.5 cm away from the cavity center: \ E = \frac{k \, q}{r^2} \ where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \mathrm{N m^2/C^2} \), \( q = 1.0 \times 10^{-7} \mathrm{C} \), and \( r = 1.5 \times 10^{-2} \mathrm{m} \).

Step 5: Calculating the electric field

Substitute the values into the formula: \ E = \frac{(8.99 \times 10^9) \times (1.0 \times 10^{-7})}{(1.5 \times 10^{-2})^2} \ Simplify to find \( E = 3.994 \times 10^{5} \mathrm{N/C} \).

Step 6: Conclusion for both parts

For part (a): The electric field at 1.5 cm from the cavity center is \( 3.994 \times 10^5 \mathrm{N/C} \). For part (b): Anywhere in the metal, the electric field is zero.

Key Concepts

Coulomb's Law and Its Application

Coulomb's law is a fundamental principle used to calculate the electric force between two charged particles. In context, if you have a charged particle at the center of a spherical cavity, Coulomb's law helps determine the electric field generated by this charge at any specific point inside the cavity. The law is expressed with the formula:\[ E = \frac{k \, q}{r^2} \]Where:

• \(E\) is the electric field strength.
• \(k\) is Coulomb's constant \(8.99 \times 10^9 \, \mathrm{N m^2/C^2}\).
• \(q\) is the charge amount \(1.0 \times 10^{-7} \, \mathrm{C}\).
• \(r\) is the distance from the charge to the point of interest.

For example, if you want to find the electric field at a point 1.5 cm from the cavity center, you substitute \(r\) into the formula. Even though the charge is confined to a small center region, it still exerts influence over the entire cavity area, and Coulomb's law captures this behavior effectively. Understanding and applying this law can simplify calculating forces and fields in various physics problems.

Electrostatic Shielding

Electrostatic shielding is an interesting phenomenon that occurs when a conductor is present in an electric field. A charged body within or near a conducting shell induces free electrons in the conductor to rearrange themselves in such a way that cancels out the effect of the external electric field. This happens because conductors allow electrons to move freely, which lets them redistribute to oppose any applied charge or field. In the exercise, the charged particle is positioned inside a metal's cavity. Outside the immediate cavity area and in the surrounding metal, all these induced charges combine to zero out the electric field. This means the electric field inside the metal, beyond the cavity's inner wall, is null, thanks to electrostatic shielding. This concept is crucial in many practical applications:

• It is applied in creating Faraday cages, which are structures used to shield sensitive electronic equipment.
• This is used in designing protective gear that prevents harm from electrostatic charges.

As shown, understanding electrostatic shielding provides insight into how conductors handle external charges and fields.

Metal Conductor Properties

Metals, used in many electrical and mechanical systems due to their distinct properties, play a crucial role in electricity dynamics. Free electrons are a feature of metal conductors, allowing them to conduct electric currents very efficiently. These free electrons can move freely inside the metal structure, which is the basis for several of the metal's unique electrical properties. When a charge is introduced inside or near a metal body, the free electrons react by moving to negate electric influence inside the conductor. This rearrangement happens quickly and effectively, making metals excellent for shielding purposes. Here are a few crucial metal conductor properties that are relevant:

• High electrical conductivity due to the movement of free electrons.
• Good heat conductors, which often go hand-in-hand with electrical conductivity.
• Effective at electrostatic shielding, as seen in the problem where the field is zero in the metal.

These characteristics explain why metals are so prevalently used in electrical circuits and protective settings, ensuring that electric fields remain controlled and contained where needed.