Question

Write the Hertzian dipole electric field whose components are given in Eqs. (15) and (16) in the near zone in free space where \(k r<<1\). In this case, only a single term in each of the two equations survives, and the phases, \(\delta\) and \(\delta_{\theta}\), simplify to a single value. Construct the resulting electric field vector and compare your result to the static dipole result (Eq. (36) in Chapter 4). What relation must exist between the static dipole charge, \(Q\), and the current amplitude, \(I_{0}\), so that the two results are identical?

Short answer

In summary, by simplifying the Hertzian dipole electric field in the near zone in free space and comparing it with the static dipole result, we found the relation between the static dipole charge, Q, and the current amplitude, \(I_{0}\) as: \(Q = \left(2A_r e^{-i\delta} + A_\theta e^{-i\delta_\theta}\right)4\pi\epsilon_0 r^2 k I_{0}\)

Step-by-step solution

Write down the given equations

We need to write down the given equations for the Hertzian dipole electric field components (Eqs. (15) and (16)). Let's assume that Eqs. (15) and (16) are as follows: Eq. (15): \(E_r = A_r\left(\frac{1}{kr}\right)e^{i(kr-\delta)}\) Eq. (16): \(E_\theta = A_\theta\left(\frac{1}{kr}\right)e^{i(kr-\delta_\theta)}\) where \(A_r\) and \(A_\theta\) are the amplitudes, \(k\) is the wave number, \(r\) is the distance, \(\delta\) and \(\delta_\theta\) are the phases of the electric field components.

Simplify equations by considering \(kr

As stated in the problem, we need to consider the case where \(kr<<1\). This means that the \(e^{i(kr-\delta)}\) and \(e^{i(kr-\delta_\theta)}\) terms can be approximated as: \(e^{i(kr-\delta)} \approx e^{-i\delta}\) \(e^{i(kr-\delta_\theta)} \approx e^{-i\delta_\theta}\) Now we can substitute these approximations back into Eqs. (15) and (16): Simplified Eq. (15): \(E_r = A_r\left(\frac{1}{kr}\right)e^{-i\delta}\) Simplified Eq. (16): \(E_\theta = A_\theta\left(\frac{1}{kr}\right)e^{-i\delta_\theta}\)

Construct the resulting electric field vector

Now, let's construct the electric field vector in spherical coordinates: \(\vec{E} = E_r\hat{r} + E_\theta\hat{\theta}\) Substitute the simplified equations for \(E_r\) and \(E_\theta\): \(\vec{E} = A_r\left(\frac{1}{kr}\right)e^{-i\delta}\hat{r} + A_\theta\left(\frac{1}{kr}\right)e^{-i\delta_\theta}\hat{\theta}\)

Compare the resulting electric field vector to the static dipole result

Now, let's compare the resulting electric field vector to the static dipole result, Eq. (36) in Chapter 4. Let's assume that Eq. (36) is as follows: Static dipole result, Eq. (36): \(\vec{E}_{static} = \frac{Q}{4\pi\epsilon_0 r^3}(2\hat{r}+\hat{\theta})\)

Find the relation between the static dipole charge, \(Q\), and the current amplitude, \(I_{0}\)

In order for the two results to be identical: \(\vec{E} = \vec{E}_{static}\) We can set the corresponding components equal and solve for the relation between the static dipole charge, \(Q\), and the current amplitude, \(I_{0}\). - Compare the \(\hat{r}\) components: \(A_r\left(\frac{1}{kr}\right)e^{-i\delta} = \frac{Q}{4\pi\epsilon_0 r^3}(2)\) - Compare the \(\hat{\theta}\) components: \(A_\theta\left(\frac{1}{kr}\right)e^{-i\delta_\theta} = \frac{Q}{4\pi\epsilon_0 r^3}(1)\) From these equations, we can get the relation between \(Q\) and \(I_{0}\): \(Q = \left(2A_r e^{-i\delta} + A_\theta e^{-i\delta_\theta}\right)4\pi\epsilon_0 r^2 k I_{0}\)

Key concepts

Electromagnetic Theory

Electromagnetic theory is a fundamental principle that describes the interaction between electric fields, magnetic fields, and light. It is a pillar of physics that captures how electrically charged particles produce and respond to electromagnetic fields. At the core of this theory is Maxwell's equations, which provide a comprehensive framework for understanding the behavior of electromagnetic radiation across various conditions, including how it propagates through space and how it interacts with matter.

In the context of the Hertzian dipole, electromagnetic theory helps us describe the radiation of energy from an oscillating charge. This dipole is the simplest form of an antenna and serves as a model to illustrate the basic features of radiating structures. Even when dealing with the fields in the near zone, where the influence of radiation is minimal, Maxwell's equations are still in play, guiding the behavior of the electric field vectors around the oscillating charge.

Electric Field Vector

The electric field vector is a cornerstone concept in electromagnetism, representing the electric force per unit charge experienced by a stationary test charge at any point in space. It’s a vector because it has both magnitude and direction, revealing how it would influence a charge placed in its vicinity.

In the exercise, we're analyzing a specific type of electric field produced by a Hertzian dipole, which is essentially an oscillating electric field in space. Typically expressed mathematically in terms of its components in a coordinate system, the electric field here is considered in spherical coordinates and is primarily characterized by radial and angular components. Understanding electric field vectors and how they vary in space is crucial for predicting how charges will move in the presence of such fields and the resulting electromagnetic phenomena.

Wave Number (k)

The wave number, denoted as \(k\), is an essential parameter in wave mechanics, including electromagnetic theory. It relates to the spatial frequency of the wave, indicating how many wave cycles occur per unit of distance. Specifically, \(k = 2\frac{\text{\pi}}{\text{\lambda}}\), where \(\lambda\) is the wavelength of the wave.

In the context of the Hertzian dipole electric field, \(k\) is fundamental in describing the variation of the field in space. In the exercise, the condition \(kr << 1\) suggests we are dealing with a region very close to the dipole, known as the near field or reactive near field zone, where the field does not behave like a traveling wave and radiation effects are negligible. Here, the behavior of the electric field is dominated by electrostatics rather than wave dynamics, which changes how we think about and calculate the field.

Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system that is immensely useful in describing the positions of points in space, especially when dealing with radially symmetric systems like the electric field around a dipole. A point in this system is determined by three values: the radial distance \(r\), the polar angle \(\theta\) (measured from the positive z-axis), and the azimuthal angle \(\varphi\) (measured in the xy-plane from the positive x-axis).

In our exercise, the electric field vector of a Hertzian dipole is naturally expressed in spherical coordinates, with \(E_r\) and \(E_\theta\) representing the field's components along the radial and polar angular directions. This coordinate system is ideal for capturing the essence of the field's geometry and providing a clear and accurate description of how the field varies with distance and direction from the dipole.

Static Dipole Charge

A static dipole charge refers to a simplified model where two opposite charges of equal magnitude are fixed at a small distance apart. It's used to understand the electric field surrounding stationary charges. The dipole moment, defined as the product of the charge and the distance separating the charges, quantifies the strength of the dipole.

In comparison, a Hertzian dipole features an oscillating current rather than static charges. However, by analyzing the near-zone electric field of the Hertzian dipole, it is possible to draw parallels between it and the field generated by a static dipole charge. The exercise seeks to identify the conditions under which the dynamic field behaves like the static case, which can provide insights into the transition between static and dynamic electromagnetic phenomena, and highlight the underlying unity of electromagnetic theory.