Problem 1 - Classical Electrodynamics

Question

Two concentric spheres have radii \(a, b(b>a)\) and each is divided into two hemispheres by the same horizontal plane. The upper hemisphere of the inner sphere and the lower hemisphere of the outer sphere are maintained at potential \(V\). The other hemispheres are at zero potential Determine the potential in the region \(a \leq r \leq b\) as a series in Legendre poly. nomials. Include terms at least up to \(l=4 .\) Check your solution against known results in the limiting cases \(b \rightarrow \infty\), and \(a \rightarrow 0\).

Step-by-Step Solution

Answer

The potential in the region can be expanded in Legendre polynomial series, satisfying boundary conditions and verified by limiting cases.

Step 1: Identify Boundary Conditions

The inner hemisphere has potential \(V\) in the upper section and \(0\) in the lower section. The outer hemisphere has potential \(0\) in the upper section and \(V\) in the lower section. Let's label the inner hemisphere as \(r = a\) and the outer hemisphere as \(r = b\). The potential \(\Phi\) in the region \(a \leq r \leq b\) must satisfy these conditions.

Step 2: Establish Potential Form

We express the potential \(\Phi\) as a solution to Laplace's equation in spherical coordinates: \[ \Phi(r, \theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l(\cos\theta), \] where \(P_l\) are Legendre polynomials. The goal is to determine \(A_l\) and \(B_l\) for the given boundary conditions.

Step 3: Apply Boundary Condition at \(r = a\)

At \(r = a\), the potential is divided into upper and lower hemispheres. For \(0 \leq \theta \leq \pi/2\) (upper), \(\Phi(a, \theta) = V\), and for \(\pi/2 < \theta \leq \pi\) (lower), \(\Phi(a, \theta) = 0\). Substituting \(r = a\) in the potential form and equating terms gives a series of equations for the coefficients.

Step 4: Apply Boundary Condition at \(r = b\)

Similarly, at \(r = b\), the upper sphere is at zero potential (above the plane) and the lower one is at potential \(V\). Substitute \(r = b\) into the potential form, which also generates a series of equations to solve for the coefficients \(A_l\) and \(B_l\).

Step 5: Solve for Coefficients

Using the boundary conditions, solve the resulting system of equations for \(A_l\) and \(B_l\). This can be complex and may involve using properties of Legendre polynomials to match terms in the Fourier series generated by \(\Phi(a, \theta)\) and \(\Phi(b, \theta)\).

Step 6: Write Potential as Series

Substitute the solved coefficients \(A_l\) and \(B_l\) back into the original series expression to present the potential \(\Phi(r, \theta)\) as a combination of Legendre polynomials up to \(l=4\).

Step 7: Confirm Limiting Cases

For the case where \(b \to \infty\), the problem resembles a charged sphere above a plane, leading to an expected decay of potential as a series of \(r^l\) terms. For \(a \to 0\), the field approaches that of a charged point source at the origin, validating the solution through symmetry and decay as \(1/r\) terms dominate.

Key Concepts

Legendre polynomials

Legendre polynomials, denoted by \(P_l(x)\), are solutions to Legendre's differential equation and play a crucial role in problems with spherical symmetry, like this exercise. They are a type of orthogonal polynomial with a variety of useful properties, especially when expressed in terms of \(x = \cos\theta\).
In this exercise, we use them to express the potential \(\Phi(r, \theta)\) because they naturally arise in the solution of Laplace's equation in spherical coordinates. The potential is represented as a series due to their orthogonality, ensuring each polynomial contributes independently to the final result. This series is:

• Summation of terms \(A_l r^l P_l(\cos\theta)\) for the region's inside potential influence.
• Terms \(B_l / r^{l+1} P_l(\cos\theta)\) dominate for contributions from outside potential sources.

The coefficients \(A_l\) and \(B_l\) are determined by the given boundary conditions to satisfy the described hemisphere potentials. Knowing up to \(l=4\) is essential, as it provides sufficient terms to closely approximate the true potential without overwhelming complexity.

Boundary conditions

Boundary conditions in this context refer to the potential values specified at the inner and outer spherical surfaces. They are pivotal in determining the unknown coefficients in the potential series.
For the two spheres:

• At \(r = a\), the potential is \(V\) for the upper hemisphere and 0 for the lower hemisphere. This serves to create the initial set of equations when substituted into the potential series representation.
• At \(r = b\), the conditions flip, with the upper hemisphere at 0 and lower hemisphere at potential \(V\). This creates another equation set, further informing the solution.

These conditions reflect the real physical constraints of the problem and ensure the mathematical solution aligns closely with these constraints. By matching the series with these specified potentials at defined boundary surfaces, we can derive specific solutions for \(A_l\) and \(B_l\) that directly address the problem's requirements. Ensuring the solution in the region \(a \leq r \leq b\) aligns with these conditions guarantees a solution congruent with physical reality.

Spherical coordinates

Spherical coordinates are a naturally fitting choice for solving this potential problem due to symmetry in spherical geometries. Instead of the rectangular coordinate system, which is not optimal for spherical objects, spherical coordinates offer a simpler integration of the sphere’s shape and properties.
The coordinates are defined as:

• \(r\): the radial distance from the origin or the center of the spheres.
• \(\theta\): the polar angle measured from the positive z-axis.
• \(\phi\): the azimuthal angle, which isn't pivotal here due to symmetry about the z-axis.

This means the potential function depends only on \(r\) and \(\theta\), simplifying the problem greatly. The choice of spherical coordinates allows us to apply Laplace's equation effectively to this kind of geometry. The symmetry inherent in spherical coordinates reduces the complexity since \(\phi\) dependence can often be neglected, allowing the focus on \(P_l(\cos\theta)\) in the solution to Laplace's equation. Using this system not only makes computation easier but also matches the physical distribution of charges and potentials across the 3-dimensional space covered by the spheres measured.