In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 MHz in opposite directions inside a narrow cavity with reflectors at both ends, causing a standing-wave pattern to occur. (a) How far apart are the nodal planes of the magnetic field?
(b) If the standing-wave pattern is determined to be in its eighth harmonic, how long is the cavity?

1.364 m

10.91 m

Explanation:

Solution:-

- We are given the frequency ( f ) of the radio waves emitted in sinusoidal wave form. f = 110 MHz.

- The wavelength of EM waves ( λ ) is determined by recalling that all EM waves travel with a speed of light ( c ) i.e 3.0 * 10^8 m/s

- Therefore,

                             λ = c / f

                             λ = ( 3.0 * 10^8 ) / ( 110 * 10^6 )

                             λ = 2.727 m    

- Recall, that for standing waves we have alternating sets of nodes and anti-nodes. These points lie on the wave defined by the wavelength ( λ ). For the radio-wave to behave sinusoidaly then the nodes exist at the points where relative displacements are zero. Hence, the path difference between nodes is:

                            Node - Node = λ / 2

                           Node - Node = 2.727 / 2

                            Node - Node = 1.364 m

The nodal planes are half-wavelength apart amounting to a path difference of 1.364 m

- The length of the cavity ( L ) can be determined from the same concepts of nodes. The end-point of the standing wave is categorized by a node and starting point as anti-node ( open cavities ).

- The standing wave patterns are categorized by the order of harmonic ( n ). This tells the number of nodes found in the pattern. Also, in the previous part we calculated the path difference between two nodes. The total length of the standing wave or in other words the total length of the cavity can be determined as follows:

                          L = n*( Node - Node )

                          L = 8* ( 1.364 )

                          L = 10.91 m

The length of the cavity and the standing wave pattern is L = 10.91 m.

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