Problem 3 - Classical Electrodynamics

Question

A thin, flat, conducting, circular disc of radius \(R\) is located in the \(x-y\) plane with its center at the origin, and is maintained at a fixed potential \(V\). With the information that the charge density on a disc at fixed potential is proportional to \(\left(R^{2}-\rho^{2}\right)^{-1 / 2}\). where \(\rho\) is the distance out from the center of the disc, (a) show that for \(r \geqslant R\) the potential is $$ \Phi(r, \theta, \phi)=\frac{2 V}{\pi} \frac{R}{r} \sum_{i=0}^{\infty} \frac{(-1)^{\prime}}{2 l+1}\left(\frac{R}{r}\right)^{2 \prime} P_{21}(\cos \theta) $$ (b) find the potential for \(r

Step-by-Step Solution

Answer

(a) Use a series involving Legendre polynomials for \( r > R \). (b) Inside the disc, \( \Phi = V \). (c) Capacitance is \( C = 2 \varepsilon_0 R \).

Step 1: Understanding Potential Outside the Disc

When the distance from the center of the disc, \( r \), is greater than the radius \( R \), the potential can be expressed in terms of Legendre polynomials. The technique involves using the boundary condition on the surface of the disc and expanding the potential in spherical coordinates. This is a standard approach when dealing with Laplace's equation in spherical coordinates, as the potential outside a charged disc simplifies to a series involving Legendre polynomials.

Step 2: Confirming Potential Formulation for r>R

The potential \( \Phi(r, \theta, \phi) \) can be expanded as a series of Legendre polynomials where for \( r > R \) it is given by:\[ \Phi(r, \theta, \phi) = \frac{2V}{\pi} \frac{R}{r} \sum_{\ell=0}^{\infty} \frac{(-1)^\ell}{2\ell+1} \left(\frac{R}{r}\right)^{2\ell} P_{2\ell}(\cos \theta) \]To derive this, apply the boundary condition and integrate over the surface of the charged disc. The symmetry of the problem allows us to consider only even \( \ell \) terms in the expansion.

Step 3: Finding the Potential Inside the Disc (r<R)

For \( r < R \), the potential inside must remain constant due to the shielding effect of the conducting disc. Such a potential satisfies Laplace's equation as there's no electric field inside a conductor at equilibrium, hence potential \( \Phi(r, \theta, \phi) = V \) for \( r < R \). This constant value maintains the condition that the potential is constant on the surface of the conducting disc.

Step 4: Calculating the Capacitance of the Disc

The capacitance \( C \) of the disc can be calculated from the surface charge density \( \sigma \), which is \[ \sigma(\rho) = \frac{Q}{2\pi R^2} \left(R^2 - \rho^2\right)^{-1/2} \]Integrating this over the surface of the disc provides the total charge \( Q \), and using the definition of capacitance \( C = \frac{Q}{V} \), we find:\[ C = 2 \varepsilon_0 R \]

Step 5: Integrating to Find Charge

To find the total charge \( Q \), integrate the charge density over the entire disc:\[ Q = \int_0^{2\pi} \int_0^R \sigma(\rho) \rho \, d\rho \, d\phi \]Substituting the expression for \( \sigma(\rho) \), perform the integration which is usually solved using a trigonometric substitution, resulting in total charge \( Q \). Then calculate the capacitance using \( C = \frac{Q}{V} \).

Key Concepts

Potential Theory

Potential theory is a branch of electromagnetism that deals with electric potentials and fields. It describes how electric potentials are influenced by charges and conductors. The basic idea is that the potential around a charged object depends on the distribution of that charge. When solving problems, especially with conductors and capacitors, we can use potential theory to determine potential distributions.
In the case of the circular disc in the exercise, potential theory helps analyze the potential both inside and outside the disc. This is crucial for understanding how the fixed potential on the disc is maintained and how it influences the surrounding space. By expressing the potential in terms of spherical coordinates and solving relevant equations, we gain a clearer picture of the energetic landscape.

Charge Density

Charge density describes how electric charge is distributed over a certain area. For the conducting disc mentioned, charge density is given as proportional to \(\left(R^{2}-\rho^{2}\right)^{-1 / 2}\), where \(R\) is the radius of the disc and \(\rho\) is the radial distance from the center.
This particular form of charge density arises because charges on a conductor redistribute themselves to maintain equilibrium with the potential. The charge density is higher towards the center since that's where the potential wants to maintain a constant value against the constraints of the disc. Understanding charge density helps in calculating total charge, which is essential for determining properties like capacitance.

Laplace's Equation

Laplace’s equation is a fundamental tool for finding potential in a region without charge. It is expressed as \(abla^2 \Phi = 0\) in electrostatics. It plays a key role because it tells us how the potential behaves in regions where charges are absent.
For the disc, Laplace's equation helps determine that the potential inside the radius \(r < R\) is constant. Since there's no charge in the inner space of the conductor, the potential doesn't vary, demonstrating the shielding effect. Solving Laplace’s equation for the exterior region \(r > R\) leads to the expansion in terms of Legendre polynomials, capturing the potential caused by the disc in the surrounding space.

Legendre Polynomials

Legendre polynomials are mathematical functions used to solve differential equations that arise in spherical coordinates. They appear in the solution to Laplace's equation when dealing with symmetrical problems like the potential outside a disc.
The potential outside the disc \(\Phi(r, \theta, \phi)\) is expanded into a series that includes Legendre polynomials, \(P_{2\ell}(\cos \theta)\). This expansion lets us express complex potentials in a more manageable form, taking advantage of symmetry and ensuring the boundary conditions are met. By using Legendre polynomials, we can focus on even \(\ell\) terms to reflect the symmetry and simplify the mathematics involved. These polynomials are instrumental in showing how the potential varies with angle and distance from the disc.