Chapter 7: Laplace's Equation And Surface Potential

Chapter 7 Laplaces Equation The Potential Produced By Surface Charge

Analyze the potential function in a spherical region with radius R, which is given by '(x, y, z) = '0(z/R) 3. Determine the volume charge density Ω(r, µ) within the region r

Consider an infinite array of identical rings with charge Q and radius R, aligned along the z-axis at planes z = 0, ±b, ±2b, etc. Use Fourier expansion methods to find the potential field everywhere in space for this configuration, and verify that the solution behaves appropriately in the limit where the cylindrical variable Ω approaches R and b.

Prove, using the orthogonality of spherical harmonics, that for a function '(r) satisfying Laplace’s equation within and on a spherical surface S of radius R centered at the origin, the following hold: (a) the surface integral of '(r) over S equals 4πR²'(0); (b) the integral of z'(r) over S equals 4π/3 R^4 ∂' / ∂z evaluated at r = 0.

Determine the volume charge density Ω and surface charge density ঠon a sphere of radius R required to produce a specific interior electric field E = °2V0 xy R^3 xÌ‚ + V0 R^3 (y² ° x²) yÌ‚ + V0 R zÌ‚, with no other charges present. Express your results in terms of trigonometric functions of µ and ¡.

Derive Green’s formula relating the normal derivatives of a potential '(r) that satisfies Laplace’s equation at an equipotential surface with principal radii of curvature R₁ and R₂, showing that @²' / @n² + 2∑ (1 / R_i) @' / @n = 0, where ∑ = ½ (1 / R₁ + 1 / R₂).

Calculate the potential '(x, y) between segmented, infinite parallel plates with conducting strips of width b, where potential is fixed at staggered values, and sketch the resulting field lines and equipotential surfaces.

For a segmented conducting circle, analyze the induced charge density on segment 3 when segment 1 has unit potential and the others are grounded, considering the potential on a coaxial grounded cylindrical shell with a radius just larger than R. Show that the cross-capacitance per unit length between segments 1 and 3 is C₁₃ = (ε₀ / ln 2), demonstrating dependence only on physical constants.

Find the fraction of charge residing on the inner surface of an infinitely long, angularly truncated conducting shell with charge per unit length ∀, considering the angular parameter p.

Analyze the potential arising between an infinite grounded plane (z=0) and an infinite conical conductor at potential V, with interior angle π/4 and a small insulating gap at the vertex. Show why the potential ’(µ)’ depends solely on µ, and explicitly determine this potential.

Near the origin, approximate the potential due to four point charges located at the corners of a square of side 2a in the plane z=0, using a series expansion in x, y, z. Deduce the non-zero terms and their signs, and sketch the electric field lines and equipotential contours near the origin.

Investigate the electrostatic problem of two semi-infinite blocks of different dielectric constants, joined at a plane, with respective potentials V and 0. Determine the potential '(x, y)' between the dielectrics, the polarization charges on the interface when dielectric constants differ slightly, and the resulting electric field lines.

Model the potential in a conducting duct with a square cross section where potential varies linearly along the sides, given potentials at the corners. Find the potential function inside the duct based on boundary conditions.

Determine the potential inside a plane capacitor with one plate being a square patch at potential V, and the other at zero potential, using Fourier integral methods. Also, evaluate the total induced charge on the top plate and sketch the electric field lines for this configuration.

Using Poisson’s integral formula, derive the potential inside a sphere when the potential is specified on the surface by summing the general series solution involving Legendre polynomials. Show how this integral formulation provides the potential everywhere inside the sphere.

For a spherical shell divided into three segments maintained at potentials V, 0, and -V, find the angle µ₀ that maximizes the uniformity of the internal electric field. Calculate the induced charge densities and the interaction forces when one shell is slightly displaced from a centered position, and derive the capacitance to second order in the offset s.

Analyze a capacitor formed by an infinite grounded plane (z=0) and a grounded conical conductor at potential V and interior angle π/4, separated by a tiny insulating gap. Determine the potential distribution and the charge densities involved.

For a spherical dielectric particle embedded in an infinite medium, find the external field assuming the internal field is known, and compute the polarization surface charge density at the interface, employing boundary conditions that relate the normal components of the displacement fields.

Determine the potential inside a coaxial two-cylinder electron lens with given boundary potentials, utilizing integral identities involving Bessel functions. Find the distribution of the electric potential throughout the configuration.

Calculate the electrostatic potential in the presence of a so-called contact potential, where one half of a grounded plane is held at zero potential and the other at V, separated at an insulating gap. Use scaling arguments to establish that the potential in the upper half-space depends only on the height variable, then solve explicitly, and interpret the pattern of electric field lines.

For a parallel plate capacitor with circular plates separated by distance 2L, with given potential boundary conditions, derive the potential distribution using a Fourier-Bessel expansion, paying special attention to the boundary conditions on the plates and the resulting field continuity issues.

Address the problem of a slightly dented spherical conductor grounded at potential zero, with small deformation parametrized by a Legendre polynomial expansion. Calculate the induced surface charge density, considering the shape perturbation and boundary conditions on the surface and potential.

Revisit the problem of a segmented cylindrical shell with specified potentials, employing separation of variables in cylindrical coordinates, and determine the potential distribution within the shell.

Finally, for a simplified two-dimensional potential problem with infinite grounded electrodes at different potentials in the plane, use Fourier transforms to derive the potential and electric field distributions, analyzing the boundary conditions and their influence on the field geometries.

References

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