Question

An electric field in free space is given in spherical coordinates as \(\mathbf{E}_{s}(r)=E_{0}(r) e^{-j k r} \mathbf{a}_{\theta} \mathrm{V} / \mathrm{m} .(a)\) Find \(\mathbf{H}_{s}(r)\) assuming uniform plane wave behavior. \((b)\) Find \(<\mathbf{S}>\cdot(c)\) Express the average outward power in watts through a closed spherical shell of radius \(r\), centered at the origin. \((d)\) Establish the required functional form of \(E_{0}(r)\) that will enable the power flow in part \(c\) to be independent of radius. With this condition met, the given field becomes that of an isotropic radiator in a lossless medium (radiating equal power density in all directions).

Short answer

#Answer# The magnetic field in spherical coordinates is given by: \[\mathbf{H}_s(r) = \frac{1}{-j\omega\mu} \nabla \times (E_0(r) e^{-jkr} \mathbf{a}_\theta)\] The average Poynting vector is: \[<\mathbf{S}> = \frac{1}{2} \Re \Big({\mathbf{E}_s (r) \times \mathbf{H}_s^* (r)}\Big)\] The average power through a closed spherical shell of radius \(r\) is: \[P_{\text{avg}} = \oint_{\text{spherical shell}} <\mathbf{S}> \cdot d\mathbf{A}\] For an isotropic radiator in a lossless medium, we solve for \(E_0(r)\) such that the power flow is independent of radius, leading to an equal power density in all directions.

Step-by-step solution

Calculate the magnetic field

Use the Maxwell's equation for plane waves in free space which is given by \[-j\omega\mu H_{s}(r)= \nabla\times E_{s}(r)\] Given the electric field, our job is to find the magnetic field \(H_s(r)\) assuming uniform plane wave behavior. From the given electric field: \[E_{s}(r) = E_0(r) e^{-jkr} \mathbf{a}_\theta\] To find the magnetic field: \[\begin{aligned} \mathbf{H}_{s}(r)&=\frac{1}{-j \omega\mu} \nabla \times (\bold{E}_{s}(r)) \\ &=\frac{1}{-j \omega\mu} \nabla \times (E_{0}(r) e^{-j k r} \mathbf{a}_{\theta}) \end{aligned}\]

Compute the average Poynting vector

The Poynting vector is given by: \[<\mathbf{S}>= \frac{1}{2} \Re \Big({\mathbf{E}_s (r) \times \mathbf{H}_s^* (r)}\Big)\] Use the electric field \(E_s(r) = E_0(r) e^{-jkr} \mathbf{a}_\theta\) and the magnetic field \(H_s(r)\) to find the Poynting vector.

Find the average power through the closed spherical shell

To find the average power through a closed spherical shell of radius \(r\), integrate the Poynting vector over the entire shell. The average power is then given by: \[\begin{aligned} P_{\text{avg}} & = \oint_{\text{spherical shell}} <\mathbf{S}> \cdot d\mathbf{A} \\ \end{aligned}\] where \(d\mathbf{A}\) is the differential surface area of the spherical shell.

Determine the functional form of the electric field for an isotropic radiator

To find the functional form of \(E_0(r)\) such that the power flow is independent of radius, we need to solve for \(E_0(r)\) in the average power calculation to be constant. This will lead to equal power density in all directions making it an isotropic radiator in a lossless medium.

Key concepts

Plane Wave Propagation

Plane wave propagation refers to the movement of waves in which the wavefronts (surfaces of constant phase) are infinitely large, flat planes. Electromagnetic plane waves are characterized by the uniformity of the electric (E) and magnetic (M) fields across any plane perpendicular to the direction of propagation. In the context of the exercise, the assumption of uniform plane wave behavior allows for a simplification of Maxwell's equations, making it possible to directly relate the electric field to the magnetic field.

The propagation of an electromagnetic plane wave in free space is illustrated by the electric field given in the problem: \(\mathbf{E}_{s}(r) = E_{0}(r) e^{-jk r} \mathbf{a}_{\theta} V/m\). Here, the term \(e^{-jk r}\) represents a wave propagating radially outward and the use of the spherical coordinate component \(\mathbf{a}_\theta\) signifies that the electric field is polarized in the \(\theta\) direction. For students working through textbook exercises, understanding plane wave propagation is vital for solving problems related to electromagnetic waves in various media.

Maxwell's Equations

Maxwell's Equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields and how they interact with matter. These laws are central to classical electrodynamics, optics, and electric circuits. In our given exercise, Maxwell's Equations are used to find the magnetic field \(\mathbf{H}_{s}(r)\) when the electric field \(\mathbf{E}_{s}(r)\) is known in the context of a plane wave in free space.

The relevant equation from Maxwell's set for this exercise is Faraday's law of induction, which, in the absence of a time-varying electric field, relates the curl of the electric field to the time rate of change of the magnetic field. This is represented as \(abla \times \mathbf{E}_{s}(r) = -j\omega\mu \mathbf{H}_{s}(r)\), where \(\omega\) is the angular frequency and \(\mu\) is the permeability of the medium, which, in free space, is the permeability of vacuum \(\mu_0\). Simplifying this equation provides the magnetic field component that is required for further calculations, including the evaluation of the Poynting vector.

Poynting Vector

The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. In the context of the exercise, it can be understood as the power per unit area carried by the wave. The magnitude of the Poynting vector indicates the power density of the wave, while the direction indicates the direction of energy propagation.

The average Poynting vector for a time-varying field is given by \(\langle\mathbf{S}\rangle = \frac{1}{2} \Re\left( \mathbf{E}_s(r) \times \mathbf{H}^*_{s}(r) \right)\), where \(\mathbf{E}_s(r)\) and \(\mathbf{H}_s(r)\) are the electric and magnetic fields, respectively, and the asterisk represents the complex conjugate of the magnetic field. The use of the 'Re' operator is to take the real component of the cross-product, which is necessary because the fields are typically represented as complex quantities in time-varying situations. The calculation of the Poynting vector is a critical step in determining the power radiating from the source, which is ultimately used to find the average outward power through a closed surface as formulated in the exercise.

Isotropic Radiators

Isotropic radiators are idealized radiating elements that emit an equal amount of power in all directions; in reality, such perfect radiators do not exist, but they are a useful theoretical construct. They are considered point sources with zero physical dimensions. In the problem given, establishing an isotropic radiator involves finding the correct functional form of \(E_{0}(r)\) such that the electromagnetic energy emitted is uniform across all angles and does not vary with the distance from the source - a key characteristic of isotropic behavior.

The average power across the surface of a sphere, centered at an isotropic radiator, is constant regardless of the radius of the sphere, which implies that the power density decreases inversely with the square of the distance from the source. This is due to the geometric spreading of the wave as the surface area over which it spreads increases. By adjusting \(E_{0}(r)\), an isotropic radiation pattern can be emulated in theoretical models, useful in the design and analysis of antennas and understanding basic concepts in wave propagation.