Phy350 2014 Assignment 4
Phy350 2014 Assignment 4
Analyze electromagnetic phenomena involving a conducting magnetic shell, capacitors with dielectric materials, solenoids with time-varying currents, finite conductivity wires, spherical capacitors with polarized media, and superpositions of electromagnetic waves, focusing on magnetic and electric fields, energy, momentum, and wave behavior under quasi-static and dynamic conditions.
This comprehensive analysis explores several advanced topics in electromagnetism, emphasizing the understanding of fields, energy, momentum, and wave phenomena through the application of Maxwell's equations and related principles. Each problem involves a different configuration, requiring specific approaches grounded in classical electromagnetic theory.
Consider a long, thick cylindrical shell characterized by inner radius \( a \) and outer radius \( b \). The shell is composed of a linear, conducting magnetic material with permeability \( \mu \). A current \( I(t) \), varying slowly with time, flows uniformly along its length. To determine the electromagnetic fields \( \mathbf{B} \), \( \mathbf{H} \), and \( \mathbf{E} \) in all space regions under the quasi-static approximation, we recognize that the slow variation enables neglecting displacement currents. The symmetry suggests that the magnetic fields are azimuthal or longitudinal depending on the current configuration.
The magnetic flux density \( \mathbf{B} \) inside the shell (for \( a < r < b \)) can be derived from Ampère's law, yielding \( \mathbf{B} = \mu \mathbf{H} \) with \( \mathbf{H} \) determined by the current distribution. Outside the shell, \( \mathbf{B} \) diminishes with distance following the Biot–Savart law. The electric field \( \mathbf{E} \) arises from the induced emf due to the changing current and the magnetic field, with \( \mathbf{E} \) related to the time derivative of \( \mathbf{B} \) through Faraday's law. The exact expressions depend on the current's temporal variation and the boundary conditions imposed by the shell structure.
A parallel-plate capacitor with radius \( R \) and separation \( d \ll R \) is filled with a dielectric material
characterized by dielectric constant \( \varepsilon_r \). An oscillating voltage \( V = V_0 \cos \omega t \) is applied across the plates, generating a time-varying electric field. Ignoring fringe fields and magnetic effects at the plates' surfaces, the electric field between the plates is uniform and given by \( \mathbf{E} = \frac{V}{d} \hat{\mathbf{z}} \). The free surface charge density \( \sigma_f \) on the plates can be derived from Gauss's law as \( \sigma_f = \varepsilon_0 \varepsilon_r E \).
The magnetic field \( \mathbf{B} \) between the plates is induced by the displacement current \( \mathbf{J}_d = \varepsilon_0 \varepsilon_r \frac{\partial \mathbf{E}}{\partial t} \). Applying Ampère-Maxwell law, the magnetic field's magnitude varies with distance from the axis and is azimuthal, following \( \mathbf{B}(r) = \frac{\mu_0 I_{d}}{2\pi r} \), where \( I_d \) is the displacement current, expressed as \( I_d = \varepsilon_0 \varepsilon_r A \frac{\partial E}{\partial t} \). The direction follows the right-hand rule around the axis.
The Poynting vector \( \mathbf{S} \), representing energy flux, at the open edges of the capacitor can be calculated as \( \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} \), and the flux obtained by integrating over the edge surface. This indicates the flow of electromagnetic energy into or out of the capacitor in the oscillating regime.
Analyzing a solenoid with \( n \) turns per unit length and radius \( R \), carrying a current \( I(t) = kt \) that increases linearly, involves computing electromotive forces and energy storage. The magnetic field \( \mathbf{B} \) inside the solenoid is \( \mathbf{B} = \mu_0 n I(t) \hat{\mathbf{z}} \). The Poynting vector \( \mathbf{S} \), which models the energy flux, points radially inward or outward depending on the direction of \( I(t) \).
The total electromagnetic energy per unit length stored in the field is \( \frac{1}{2} L I^2 \), with \( L \) the per-unit-length inductance \( L = \mu_0 n^2 \pi R^2 \). Differentiating this energy with respect to time confirms that the power entering the volume matches \( dW/dt = d/dt \left(\frac{1}{2} L I^2 \right) \), consistent with the Poynting flux concept.
A long, straight wire of radius \( a \), conductivity \( \sigma \), and a uniform current density \( \mathbf{J} \) carries a steady current. The magnetic and electric fields at the surface can be obtained via Maxwell’s
equations, with the dominant electric field component related to resistive effects. The Poynting vector \( \mathbf{S} \) at the surface points radially inward, indicating energy flow from the electromagnetic fields into the resistor to sustain the current. The magnitude is related to the fields via \( |\mathbf{S}| = \frac{1}{\mu_0} |\mathbf{E} \times \mathbf{B}| \), with explicit expressions derived based on the conductivity and current density.
The system involves an inner sphere of radius \( a \) and an outer sphere of radius \( b \), with charge \( +Q \) and \( -Q \). The inner sphere has a uniform polarization \( \mathbf{P} \) in a given direction, leading to bound charges on its surface. The fields in the space between the spheres contain contributions from free charges and polarization. The energy stored in the fields can be calculated via the integral of the energy density \( u = \frac{1}{2} (\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}) \). The angular momentum stored in the fields arises from the cross-product of electric and magnetic fields, especially if magnetic materials or currents are involved. As the charge drains, the fields diminish, leading to a decrease in stored energy and angular momentum, with the electromagnetic momentum and angular momentum transferred to the environment or dissipated.
When two electromagnetic waves of equal amplitude \( E_0 \), traveling in opposite directions with the same wavenumber \( k \) and frequency \( \omega \), superimpose, a standing wave is formed. The electric fields are described as \( \mathbf{E}_1 = E_0 \cos(kz - \omega t) \hat{\mathbf{x}} \) and \( \mathbf{E}_2 = E_0 \cos(-kz - \omega t) \hat{\mathbf{x}} \). Their sum results in a spatially stationary pattern with nodes and antinodes, expressed as \( \mathbf{E} = 2 E_0 \cos(kz) \cos(\omega t) \hat{\mathbf{x}} \). The magnetic field components follow from Maxwell's equations, with phase relationships ensuring \( \mathbf{E} \) and \( \mathbf{B} \) are in phase for propagating waves. In a standing wave, the \( \mathbf{E} \) and \( \mathbf{B} \) fields are spatially out of phase in certain regions, but at antinodes, they are in phase, forming a fixed pattern.
The average energy density over a cycle can be calculated via \( u = \frac{\varepsilon_0}{2} E^2 + \frac{1}{2 \mu_0} B^2 \). The time-averaged Poynting vector \( \left< \mathbf{S} \right> \) averages to zero in a standing wave, indicating no net energy flux, but the instantaneous \( \mathbf{S} \) varies in space and time, with energy oscillating locally.
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