Consider a small, thin loop in the [itex](x,y)[/itex] plane centered in the origin and with radius [itex]a[/itex]. We are interested in the vector potential [itex]\mathbf{A}[/itex] generated by the loop at a point [itex]P(r, \theta, \phi)[/itex], with [itex]2 \pi a \ll r[/itex], so at a great distance (moreover, [itex]a \ll \lambda[/itex]).(adsbygoogle = window.adsbygoogle || []).push({});

We need two coordinates systems:

- one for the source point, for which variables and unit vectors will be primed ([itex]r', \theta', \phi'[/itex]);

- one for the observation point [itex]P(r,\theta,\phi)[/itex].

Because of the use of the spherical coordinates, the unit vectors do not coincide in any case.

A constant current [itex]I[/itex] flows through the loop and in any point we have only [itex]\mathbf{I} = I_{\phi} \mathbf{u}_{\phi'}[/itex], where [itex]\mathbf{u}_{\phi'}[/itex] is the unit vector in the [itex]\phi[/itex] direction.

The vector potential [itex]\mathbf{A}[/itex] is computed as usual through the integral

[itex]\mathbf{A} = \displaystyle \frac{\mu}{4 \pi} \int_{loop} \mathbf{I} \frac{e^{-j \beta R}}{R} dl'[/itex]

but we have to express every [itex]\displaystyle \mathbf{I} \frac{e^{-j \beta R}}{R} dl'[/itex] in terms of the [itex](\mathbf{u}_r, \mathbf{u}_{\theta}, \mathbf{u}_{\phi})[/itex] unit vector, so the unprimed coordinate system. We will obtain a new vector with (in general) all the three components. [itex]R[/itex] is the distance between the position of the actual length element [itex]dl'[/itex] and [itex]P[/itex]: [itex]R = |\mathbf{r} - \mathbf{r'}|[/itex].

But why the resulting [itex]\mathbf{A}[/itex] has only the [itex]\phi[/itex]-component?

So, the question is: why a vector like [itex]\mathbf{I}[/itex], which has in its own primed coordinate system only a [itex]\phi'[/itex]-component, after the shown integration originates a vector potential [itex]\mathbf{A}[/itex] with only the same component, [itex]A_{\phi}[/itex]?

This is the procedure followed in Balanis, Antenna Theory, Ch. 5.

Thank you anyway!

Emily

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Vector potential components in small loop antenna

**Physics Forums | Science Articles, Homework Help, Discussion**