An antenna is placed in a dielectric medium of dielectric constant $$6.25$$. If the maximum size of that antenna is $$5.0$$ mm, it can radiate a signal of minimum frequency of ______ GHz.
(Given $$\mu_r = 1$$ for dielectric medium)

An antenna is placed in a dielectric medium with dielectric constant $$\kappa = 6.25$$ and $$\mu_r = 1$$. The maximum size of the antenna is $$5.0$$ mm. We need to find the minimum frequency it can radiate in GHz.

Recall the antenna condition. For a quarter-wave antenna, the minimum antenna length required is one-fourth of the wavelength:

$$l = \frac{\lambda}{4}$$

For the minimum frequency (longest wavelength), the antenna size equals $$\frac{\lambda}{4}$$:

$$\lambda = 4l = 4 \times 5.0 \text{ mm} = 20 \text{ mm} = 0.02 \text{ m}$$

Find the speed of electromagnetic waves in the dielectric medium. The speed in a medium with dielectric constant $$\kappa$$ and relative permeability $$\mu_r$$ is:

$$v = \frac{c}{\sqrt{\kappa \cdot \mu_r}}$$

Substituting the given values:

$$v = \frac{c}{\sqrt{6.25 \times 1}} = \frac{c}{\sqrt{6.25}} = \frac{c}{2.5}$$

$$v = \frac{3 \times 10^8}{2.5} = 1.2 \times 10^8 \text{ m/s}$$

Calculate the minimum frequency using $$v = f\lambda$$:

$$f_{min} = \frac{v}{\lambda} = \frac{1.2 \times 10^8}{0.02}$$

$$f_{min} = 6 \times 10^9 \text{ Hz} = 6 \text{ GHz}$$

The minimum frequency of the signal the antenna can radiate is 6 GHz.