Question

Consider an elliptical conducting thread with a positive charge. Although the thread is thin, its stiffness prevents Coulomb's forces from deforming its elliptical shape. There exists a specific linear charge density (λ) for the charge distribution on this thread, which, unlike a charged plate or sphere, doesn't necessarily nullify the electric field inside the closed thread.

Due to Coulomb's force, there should be a unique λ as a function of θ, where θ is an arbitrary angle relative to the horizontal at the ellipse's center. The ellipse's polar form relative to the center is given by:
r(θ)=b/√(1−(ecosθ)²)

I want to know:

Can we apply the Poisson equation to determine λ(θ)?
If yes, how can we replace the surface charge density in the Poisson equation with linear charge density? Is it easy to solve the relevant Poisson equation for this example? What is the solution?

If the Poisson equation is not applicable to this example, how can we obtain λ(θ)?

Answer 1

To determine the linear charge density λ(θ) on an elliptical conducting thread, one cannot directly apply the Poisson equation due to the complex geometry and the linear nature of the charge density. Instead, Gauss's law and Coulomb's law could be used, although solving for λ(θ) would likely require advanced numerical or analytical methods.

Step-by-step explanation:

The question involves determining the linear charge density λ(θ) on an elliptical conducting thread using electric field and charge distribution concepts from electromagnetism. While the Poisson equation is typically used to find electric potential distributions for continuous charge distributions, it's not easily applicable to this scenario primarily due to the complex geometry of the ellipse and the linear charge density we are dealing with. To obtain λ(θ), one would need to start from Coulomb's law and consider the symmetry and boundary conditions specific to the ellipse.

For a conducting surface, Gauss's law is often used where the electric field inside is zero and only the outer surface contributes to the flux. However, converting surface charge density to linear charge density might involve integrating charge elements over the curve, which is not straightforward for non-uniform distributions like an elliptical thread with variable charge density.

Considering Gauss's law, the integral form applied to the charge on the elliptical thread would require exploring the specific geometry of the ellipse. The fact that electric field only exists outside the thread suggests a potential approach to solving for λ(θ) could be through the superposition of field contributions from differential charge elements around the ellipse. Due to the complexity, this is not a typical textbook problem and would likely require numerical methods or specialized analytical techniques to solve for the exact function of λ(θ).