Question

For radio and microwaves, the depth of penetration into the human body is proportional to ${\mathrm{\lambda }}^{1/2}$. If$27\mathrm{MHz}$radio waves penetrate to a depth of$14\mathrm{cm}$, how far do$2.4\mathrm{GHz}$microwaves penetrate$?$

Short answer

Microwaves penetrates at$1.53\mathrm{cm}$.

Step-by-step solution

Calculation for wavelength for first frequency

The depth of penetration into the human body for radio and microwaves is provided by ${\mathrm{\lambda }}^{1/2}$, which is important for our situation.

Formula is,

$\mathrm{c}=\mathrm{\lambda }\times \mathrm{f}$

For $\mathrm{f}=27\mathrm{MHz}$

${\mathrm{\lambda }}_1=\frac{\mathrm{c}}{\mathrm{f}}$

${\mathrm{\lambda }}_1=\frac{3\times {10}^8{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}}{27\times {10}^6}$

${\mathrm{\lambda }}_1=11\mathrm{m}$

$\sqrt{{\mathrm{\lambda }}_1}=3.2\mathrm{m}$

Calculation of wavelength for second frequency

For $\mathrm{f}=2.4\mathrm{GHz}$

${\mathrm{\lambda }}_2=\frac{\mathrm{c}}{\mathrm{f}}$

${\mathrm{\lambda }}_2=\frac{3\times {10}^8{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}}{2.4\times {10}^9}$

${\mathrm{\lambda }}_2=0.125\mathrm{m}$

Calculation for distance

The relationship is,

$\mathrm{r}=\frac{\sqrt{{\mathrm{\lambda }}_1}}{\sqrt{{\mathrm{\lambda }}_2}}$

$=\frac{3.2\mathrm{m}}{0.35\mathrm{m}}$

$=9.14$

Depth of penetration is,

${\mathrm{d}}_2=\frac{{\mathrm{d}}_1}{\mathrm{r}}$

${\mathrm{d}}_2=\frac{14\mathrm{cm}}{9.14}$

${\mathrm{d}}_2=1.53\mathrm{cm}$