1.5 Other Charge Distributions – Fields & Potentials

Other Charge Distributions Fields & Potentials

Advanced Charge Distributions: Fields & Potentials

Introduction to Extended Charge Distributions

In many real-world scenarios, charges are distributed across extended objects such as rings, lines, or sheets rather than concentrated at a single point. In this guide, we’ll explore how to calculate electric fields and potentials for these complex distributions using integrals and symmetry principles. If you need a refresher, revisit earlier sections on Gauss’s Law and Electric Potentials.

Calculating Electric Fields for Extended Charge Distributions

For extended charge distributions, the total charge is divided into infinitesimal elements, each contributing a small electric field. By integrating these contributions, we can determine the total field for the distribution.

General Formula:

Where:

• : Coulomb’s constant.

• : Infinitesimal charge element.

• : Distance from the charge element to the point of interest.

•  : Unit vector in the direction of the field.

Examples of Charge Distributions

1. Ring of Charge

A ring with a total charge and radius creates an electric field along its axis. By symmetry, the horizontal components of the field cancel, leaving only the vertical component.

The resulting field at a distance from the center along the axis is:

Special Case:

When , the field simplifies to:

This matches the field for a point charge, demonstrating that a distant ring behaves like a point charge.

2. Line of Charge

For a uniformly charged line with linear charge density , the electric field at a point is determined by integrating along the length of the line.

Using the Pythagorean theorem to express the distance :

The resulting field is:

3. Sheet of Charge

For an infinite sheet of charge with surface charge density , the electric field is uniform and given by:

Where is the permittivity of free space.

Gauss’s Law for Various Shapes

Using Gauss’s Law, we can calculate electric fields for symmetrical charge distributions efficiently.

Line of Charge:

For a charged line enclosed in a cylindrical Gaussian surface:

Sphere:

For a charged sphere (radius):

1. Outside the Sphere:

2. Inside the Sphere:

Insulating Sheet:

For a uniformly charged insulating sheet:

Potential Difference for Various Shapes

The potential difference can be calculated by integrating the electric field along a path:

Examples:

1. Line of Charge:

2. Conducting Sheet: The potential difference between the sheet and a point away is:

Practice Problems

Problem 1: Electric Field Inside a Shell

Question: A conducting spherical shell with inner radius and outer radius encloses a charge . What is the field at a point between and ?

Answer: Using Gauss’s Law, the field depends only on , since the charge is on the shell’s surface. The field is:

Problem 2: Potential Difference for a Line of Charge

Question: Calculate the potential at a distance from an infinite line of charge with linear charge density .

Answer: Using symmetry and integrating:

Problem 3: Field Inside a Sphere

Question: For a uniformly charged sphere with total charge , find the field inside the sphere.

Answer: Using Gauss’s Law:

Advanced Practice: Free Response Question

Question:

A non-conducting solid sphere of radius contains a charge distributed uniformly. It is enclosed by a spherical shell with inner radius and outer radius , containing a charge . Express all answers in terms of , , and constants.

1. Derive the electric field in the following regions:

2. Calculate the potential difference between and.