A long straight line of uniform charge density

c/m

is surrounded by an insulating

medium out to a radius Rm. Find

D

: Also find the electric field in the

region outside the insulation. Explain why the electric field cannot be found

in the insulation region.

To tackle this problem, we need to understand the behavior of electric fields around charged objects, particularly in the context of a long straight line of uniform charge density. Let's break this down step by step.

Understanding Charge Density and Electric Displacement Field (D)

First, let's define the charge density. If we have a long straight line with a uniform charge density denoted by λ (lambda, in coulombs per meter), we can express the electric displacement field, D, in the surrounding medium. The electric displacement field is related to the free charge density and the permittivity of the medium.

Calculating D in the Surrounding Medium

For a long straight line of charge, the electric displacement field D at a distance r from the line can be calculated using Gauss's law for electric displacement:

• The formula for D is given by: D = λ / (2πr)
• Here, λ is the linear charge density, and r is the radial distance from the line of charge.

So, outside the insulating medium (beyond radius R), the electric displacement field D can be expressed as:

D = λ / (2πr), where r > R.

Electric Field Outside the Insulation

Next, we need to find the electric field E in the region outside the insulation. The relationship between the electric field E and the electric displacement field D is given by:

E = D / ε, where ε is the permittivity of the medium.

Substituting the expression for D, we get:

E = λ / (2πεr).

This equation shows that the electric field decreases with distance from the line of charge, which is a characteristic behavior of electric fields around linear charge distributions.

Why the Electric Field Cannot Be Found in the Insulation Region

Now, let’s address why we cannot find the electric field in the insulating region itself. The key point here is that in a perfect insulator, there are no free charges available to respond to the electric field. This means that:

• The electric field inside an ideal insulator is zero because the charges cannot move to create a field.
• In the insulating material, any induced charges would be bound and unable to move freely.

Thus, while the electric field exists outside the insulating medium, it does not penetrate into the insulating material. This is a fundamental property of dielectrics and insulators, which is crucial in understanding how electric fields behave in different materials.

Summary of Key Points

To summarize:

• The electric displacement field D outside the insulation is given by D = λ / (2πr).
• The electric field E outside the insulation is E = λ / (2πεr).
• Inside the insulating region, the electric field is zero due to the lack of free charge movement.

This understanding of electric fields and displacement fields is essential in various applications, including capacitors, insulators, and understanding electrostatics in general. If you have any further questions or need clarification on any point, feel free to ask!