Half-Wavelength Dipole Radiation Pattern Fundamentals

A half-wavelength dipole antenna is a fundamental antenna type widely used in various applications. Its radiation pattern is distinct and understanding it is crucial for antenna design and analysis.

The radiation pattern of a half-wavelength dipole is characterized by a "doughnut" or toroidal shape. This means that the radiation is strongest in directions perpendicular to the antenna's axis and weakest (nulls) along the antenna's axis itself.

• Maximum Radiation: Occurs in the plane perpendicular to the dipole's axis.
• Nulls (Minimum Radiation): Occur along the dipole's axis.

Dipole Antenna Orientation Analysis

The question specifies that the half-wavelength dipole is kept in the x-y plane and oriented along 45° from the x-axis. Let's break down what this means in terms of spherical coordinates ($\theta$, $\Phi$):

• The dipole is in the x-y plane. In spherical coordinates, any point in the x-y plane corresponds to an angle $\theta = 90^\circ$ (measured from the z-axis).
• The dipole is oriented along 45° from the x-axis. This means its axis lies along the line where $\Phi = 45^\circ$ (measured from the x-axis in the x-y plane).
• Therefore, the axis of the dipole antenna is defined by the direction where $\theta = 90^\circ$ and $\Phi = 45^\circ$.

Determining Null Directions in the Radiation Pattern

As established, the nulls of a half-wavelength dipole radiation pattern occur precisely along the antenna's axis.

Given that the dipole's axis is oriented at:

• $\theta = 90^\circ$
• $\Phi = 45^\circ$

Then, the direction of a null in the radiation pattern will also be along this specific orientation.

The far-field electric field strength of a half-wavelength dipole is proportional to $\sin(\Psi)$, where $\Psi$ is the angle measured from the antenna's axis. Nulls occur when $\sin(\Psi) = 0$, which means $\Psi = 0$ or $\Psi = \pi$. These angles correspond to directions *along* the antenna's axis.

Since the question asks for the null direction for $0 \le \Phi \le \pi$, the relevant null direction is $(\theta = 90^\circ, \Phi = 45^\circ)$. The other null would be at $(\theta = 90^\circ, \Phi = 225^\circ)$, which is outside the specified range for $\Phi$.

Summary of Null Direction

Based on the antenna's orientation and the characteristics of a half-wavelength dipole's radiation pattern, the null direction is where the observer is directly in line with the antenna's length.

For an antenna in the x-y plane oriented at $\Phi = 45^\circ$, the null direction is indeed:

$\theta = 90^\circ$, $\Phi = 45^\circ$

This corresponds to the direction where the dipole antenna itself is physically located in the x-y plane, 45 degrees from the x-axis.