Question

Two microwave signals of nearly equal wavelengths can gener- ate a beat frequency if both are directed onto the same microwave detector. In an experiment, the beat frequency is 100 MHz. One microwave generator is set to emit microwaves with a wavelength of 1.250 cm. If the second generator emits the longer wavelength, what is that wavelength

Answer 1

the longer wavelength is 1.2552 cm

Step-by-step explanation:

given that

beat frequency
`f_(b)` = 100 MHz = 100 × 10⁶ Hz

λ₁ = 1.250 cm = 0.0125 m

we know that beat frequency
`f_(b)` of two simultaneous frequencies f₁ and f₂ is expressed as;

`f_(b)` = | f₁ - f₂ |

we know that microwave travels at a speed of light, so for any electromagnetic wave traveling at speed of light c with wavelength λ; frequency is;

f = c / λ

hence our beat frequency
`f_(b)` becomes

`f_(b)` = c ( 1/λ₁ - 1/λ₂)

to find the longer wavelength, λ₂

`f_(b)` = c ( 1/λ₁ - 1/λ₂)

divide both side by c

`f_(b)` /c = ( 1/λ₁ - 1/λ₂)

1/λ₂ = 1/λ₁ -
`f_(b)` /c

λ₂ = [1/λ₁ -
`f_(b)` /c ]⁻¹

so we substitute in our values

we know that speed of light c = 3 × 10⁸

so

λ₂ = [ (1/0.0125) - (100 × 10⁶ /3 × 10⁸) ]⁻¹

λ₂ = [80 - 0.3333 ]⁻¹

λ₂ = [79.6667 ]⁻¹

λ₂ = 0.01255 m

λ₂ = 0.012552 × 100 cm

λ₂ = 1.2552 cm

Therefore, the longer wavelength is 1.2552 cm