Electric Field of Coaxial Conducting Cylinders

How to calculate the resultant differential force due to the field of the inner cylinder acting on the outer cylinder?

What are the specific steps and formulas involved in determining the differential force in this scenario?

Calculating the Resultant Differential Force

To calculate the resultant differential force due to the field of the inner cylinder acting on the outer cylinder, we can follow the steps below:

Step 1: Determine the Current and Radius Values

Identify the current values for the outer and inner cylinders, which are -100az A/m and +500az A/m respectively. Also, note the radii of the cylinders, which are 5 mm and 1 mm for the outer and inner cylinders respectively.

Step 2: Calculate the Differential Force (dF)

Use the formula dF = K_outer × B to find the resultant differential force. Here, K_outer represents the surface current density of the outer conducting cylinder, and B is the magnetic field density acting on the outer cylinder due to the inner cylinder.

Step 3: Integrate the Differential Force

Integrate the y-component of the differential force over the upper half of the cylinder from 0 to π in the ϕ direction and from 0 to 1 in the z direction. The integral expression is F_y = ∫₀¹ ∫₀^π dF⋅(rho_outer * a_y) dϕ dz.

When dealing with the electric field of coaxial conducting cylinders, it is essential to understand how to calculate the resultant differential force acting on the outer cylinder due to the field of the inner cylinder. By following a step-by-step approach and utilizing the provided formulas, we can determine the necessary force to split the outer cylinder apart along its length.

Firstly, identifying the current values and radii of the cylinders is crucial in setting up the calculations. Once these values are known, we can proceed to calculate the differential force using the formula dF = K_outer × B. This formula takes into account the surface current density of the outer conducting cylinder and the magnetic field density acting on the outer cylinder.

Integrating the differential force over the specified range in the cylindrical coordinates further refines our understanding of the electric field interaction between the coaxial cylinders. By following the integral expression F_y = ∫₀¹ ∫₀^π dF⋅(rho_outer * a_y) dϕ dz, we can determine the y-component of the resultant force accurately.