Question

(a) Two microwave frequencies are authorized for use in microwave ovens: 900 and 2560 MHz. Calculate the wavelength of each. (b) Which frequency would produce smaller hot spots in foods due to interference effects?

Short answer

(a) The required wavelengths are $3.33\times {10}^{0}m$and $1.33\times {10}^{-1}m$.

(b) The frequency that would produce smaller hot spots in food is

Step-by-step solution

Definition of wavelength and frequency

The distance between two successive crests or troughs of a wave is defined as its wavelength. It is measured in the wave's direction.

The frequency of a repeating event is the number of occurrences per unit of time. It is also known as temporal frequency to distinguish it from spatial frequency, and ordinary frequency to distinguish it from angular frequency.

Given data

The frequency is$f_1=900MHz\left(\frac{{10}^{6}Hz}{1MHz}\right)=9.00\times {10}^{8}Hz$

The frequency is$f_2=2560MHz\left(\frac{{10}^{6}Hz}{1MHz}\right)=2.56\times {10}^{6}Hz$

Calculate the wavelengths of the frequency (a)

The relation between the wavelength and frequency f can be expressed as,

$\lambda =\frac{c}{f}$……………..(1)

Where c is the propagation speed of light, i.e, $c=3\times {10}^{8}m/s$.

For the frequency $f_1$, we can get the value of wavelength using equation (1), such that,

$$\begin{gathered} \lambda =\frac{c}{f_1} \\ =\frac{3\times {10}^{8}m/s}{9.00\times {10}^{8}Hz} \\ =0.34m \end{gathered}$$

For the frequency $f_2$, we can get the value of wavelength using equation (1), such that,

$$\begin{gathered} {\lambda }_2=\frac{c}{f_2} \\ =\frac{3\times {10}^{8}m/s}{2.56\times {10}^{9}Hz} \\ =0.12m \end{gathered}$$

Therefore, the required wavelengths are 0.34 m and 0.12 m.

Step 2: Finding the frequency that would produce smaller hot spots in foods (b)

The wave of greater wavelength ${\lambda }_1$would produce an interference pattern of

• fewer antinodes
• wider antinodes

The wave of smaller wavelength ${\lambda }_2$would produce an interference pattern of

• more antinodes
• narrower antinodes

The wave with smaller hotspots (antinodes) in the interference pattern is the one with frequency $f_2$.