PHY101 Lectures 23-45

Lecture 23 · Electrostatics II · Introduction to Charges:

In the last lecture, we discussed the fundamental concept of charges. Charges are entities that exert forces on each other. These forces are governed by certain principles. One key fact is that charge is conserved, meaning the total amount of charge remains constant in any closed system.

Coulomb's Law:

Coulomb's Law describes the force between two-point charges. If there are two charges, 1q and q , separated by a distance r , the force F between them is given by:

q q F k r

where k is Coulomb's constant. This force is a vector quantity, meaning it has both magnitude and direction. The direction of the force is along the line connecting the two charges. Importantly, if charge 1q exerts a force on charge q , charge q exerts an equal and opposite force on charge 1q .

Electric Fields:

Previously we have discussed the concept of electric fields. When a charge is present in an electric field, it experiences a force. The strength of this force depends on the strength of the field. If a charge Q is placed in an electric field E , the force F acting on the charge is:

F Q E

Continuous Charge Distributions:

In previous studies, the focus has been on point charges, which are idealized models where the entire charge is assumed to be concentrated at a single point. However, in many real-world scenarios, charges are distributed over a larger region of space rather than being concentrated at a single point. Understanding how to deal with these continuous charge distributions is crucial for accurately calculating the electric fields they produce.

Summing Up Contributions from Small Elements:

When dealing with continuous charge distributions, the electric field at any point in space is the result of contributions from all infinitesimal charge elements within the distribution. This is expressed as:

n E E E E E

Here, n E represents the infinitesimal electric field contribution from the n-th small charge element. Summing these contributions gives the total electric field E .

Transition to Integration:

Since the charge distribution is continuous, the sum of the contributions from all infinitesimal charge elements is represented by an integral:

n E E

This summation can be approximated by an integral for continuous distributions:

E dE

Vector Components of the Electric Field:

The electric field E is a vector quantity, meaning it has both magnitude and direction. It can be decomposed into its Cartesian components along the x, y, and z axes:

ˆ ˆ ˆ x y z E E i E j E k

Where:

x E , y E and z E are the components of the electric field in the x, y, and z directions, respectively. ˆ , , and i j k are the unit vectors in the x, y, and z directions.

Calculating Each Component:

To find the total electric field, you need to integrate the contributions to each component over the entire charge distribution:

These integrals take into account the continuous distribution of charge and calculate the contributions to the electric field in each direction.

Linear Charge Distribution:

Consider a line of charge where charges are continuously distributed along its length. To find the electric field at a point P due to this line of charge, we break the line into small segments and sum the contributions of each segment.

Suppose we have a line of charge with charge density , which is charge per unit length. If we take a small segment of length ds , the charge dq on this segment is:

dq ds

By summing the contributions of all these small segments, we can determine the total electric field at point P .

Surface and Volume Charge Distributions:

In two dimensions, we consider surface charge distributions. Here, charges are spread over a surface, such as a sheet of paper. The charge density  is charge per unit area. For a small area element dA , the charge dq is:

dq dA x x y y z z E dE E dE E dE