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A y-directed Hertzian dipole antenna having length dl, carrying a constant current I is radiating in free space.

This problem is asking for the azimuthal component of the magnetic vector potential \(\mathbf{A}_\phi\) for a Y-directed Hertzian dipole antenna.

### Problem Recap:
- A Y-directed Hertzian dipole antenna with length \(dl\) and current \(I\) is radiating in free space.
- We need to identify the correct expression for the azimuthal component (\(\mathbf{A}_\phi\)) of the magnetic vector potential.

### Relevant Theory:
For a small Hertzian dipole, the magnetic vector potential in spherical coordinates can have components in the \(\phi\) direction (azimuthal) and the \(\theta\) direction (elevation).

For a Y-directed dipole, the azimuthal component \(\mathbf{A}_\phi\) is generally derived from the vector potential expression and relates to the geometry and orientation of the dipole. The expression for \(\mathbf{A}_\phi\) typically involves a sine term corresponding to the angle \(\theta\) (the elevation angle).

The general form of the azimuthal component is given by:

\[
\mathbf{A}_\phi = \frac{\mu_0 I dl \sin\theta}{4\pi r} e^{-j\beta r}
\]

Here, \(\sin\theta\) accounts for the component of the current in the direction perpendicular to the radial vector.

### Answer Identification:
Based on the provided options, the correct answer that matches this standard result is option **d)**:

\[
\mathbf{A}_\phi = \frac{\mu_0 I dl \sin\theta}{4\pi r} e^{-j\beta r}
\]

### Final Answer:
- The correct option is **d)** \(\mathbf{A}_\phi = \frac{\mu_0 I dl \sin\theta}{4\pi r} e^{-j\beta r}\).

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Would you like more details on this problem or any further questions? Here are five related questions to consider:

1. How does the angle \(\theta\) affect the magnitude of the azimuthal component \(\mathbf{A}_\phi\)?
2. What are the differences between the azimuthal and polar components of the magnetic vector potential?
3. How is the far-field approximation applied when calculating \(\mathbf{A}_\phi\)?
4. What is the physical significance of \(\mathbf{A}_\phi\) in relation to the radiated fields of the antenna?
5. How does the distance \(r\) from the dipole influence the value of \(\mathbf{A}_\phi\)?

**Tip:** Understanding the different components of the magnetic vector potential is essential for analyzing the radiation patterns of dipole antennas.

Explore the azimuthal component of the magnetic vector potential \(\mathbf{A}_\phi\) for a Y-directed Hertzian dipole antenna radiating in free space. Understand the formula \(\mathbf{A}_\phi = \frac{\mu_0 I dl \sin\theta}{4\pi r} e^{-j\beta r}\) and its significance in electromagnetism and antenna theory.

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