How to solve the following problem:

Very large rod, with constant longitudinal charge density Q’, is set in vacuum parallel to very large conductive ribbon with width ‘a’ , where height between them is ‘a/2’ . If the surface charge density of a ribbon is ‘σ‘, calculate longitudinal force on a rod.

Thanks for reply.

To tackle the problem of calculating the longitudinal force on a charged rod positioned parallel to a conductive ribbon, we need to break it down into manageable steps. This involves understanding the electric field generated by the conductive ribbon and how it interacts with the charged rod. Let's go through the process step by step.

Understanding the Setup

We have a long rod with a uniform linear charge density denoted as Q’ and a conductive ribbon with a surface charge density of σ. The distance between the rod and the ribbon is a/2, and the width of the ribbon is a. The goal is to find the force acting on the rod due to the electric field created by the ribbon.

Step 1: Electric Field Due to the Conductive Ribbon

First, we need to determine the electric field produced by the conductive ribbon. For an infinitely wide conductive sheet with a uniform surface charge density σ, the electric field (E) created in the region above the sheet can be calculated using the formula:

• E = σ / (2ε₀)

Here, ε₀ is the permittivity of free space, approximately equal to 8.85 x 10⁻¹² C²/(N·m²). The electric field points away from the ribbon if the charge is positive and towards the ribbon if the charge is negative.

Step 2: Electric Field at the Location of the Rod

Since the rod is positioned at a distance of a/2 from the ribbon, the electric field at the location of the rod will be:

• E = σ / (2ε₀)

This electric field is uniform across the length of the rod because the rod is very long compared to the distance between it and the ribbon.

Step 3: Force on the Charged Rod

The force (F) acting on the rod can be calculated using the formula:

• F = Q * E

In this case, Q represents the total charge on the rod. Since the rod has a linear charge density Q’, the total charge can be expressed as:

• Q = Q’ * L

Here, L is the length of the rod. Substituting this into the force equation gives:

• F = (Q’ * L) * (σ / (2ε₀))

Step 4: Final Expression for the Force

Now, we can simplify the expression for the force acting on the rod:

• F = (Q’ * σ * L) / (2ε₀)

This equation shows that the force on the rod is directly proportional to both the linear charge density of the rod and the surface charge density of the ribbon, as well as the length of the rod.

Summary of the Calculation

To summarize, the longitudinal force on the rod due to the electric field created by the conductive ribbon can be calculated using the derived formula:

• F = (Q’ * σ * L) / (2ε₀)

This approach highlights the interaction between the charged rod and the electric field generated by the conductive ribbon, allowing us to quantify the force acting on the rod in a clear and logical manner.