Electromagnetic waves can form standing waves. In a standing wave pattern formed from microwaves, the distance between a node and an adjacent antinode is 0.63 cm. What is the microwave frequency?
any help is much appreciated
Well, aren't microwaves just "waving" from one end to the other? I guess you could say they like to multitask and form standing waves too! Anyway, let's get crackling with the math.
The distance between a node and an adjacent antinode in a standing wave pattern is half a wavelength (λ/2). In this case, it's given as 0.63 cm. So, we can say:
λ/2 = 0.63 cm
To find the wavelength (λ), we need to multiply both sides of the equation by 2:
λ = 2 * 0.63 cm
λ = 1.26 cm
Now, we can use the relationship between wavelength, speed, and frequency:
c = f * λ
where c is the speed of light in a vacuum (approximately 3 x 10^8 m/s), f is the frequency, and λ is the wavelength.
However, since the distance is given in centimeters, we need to convert it to meters:
λ = 1.26 cm = 0.0126 m
Plugging in the values, we have:
3 x 10^8 m/s = f * 0.0126 m
Now, let's solve for f:
f = (3 x 10^8 m/s) / (0.0126 m)
And after doing the math, we get:
f ≈ 2.38 x 10^10 Hz
So, the microwave frequency is approximately 2.38 x 10^10 Hz. Enjoy your microwaving adventures!
To find the microwave frequency in a standing wave pattern, we can use the formula:
λ = 2L/n
Where λ is the wavelength, L is the distance between the node and adjacent antinode, and n is the number of half-wavelengths within that distance.
In this case, the distance between node and adjacent antinode (L) is given as 0.63 cm. We need to convert this to meters to match the SI unit of wavelength.
1 cm = 0.01 meters
So, L = 0.63 cm * 0.01 = 0.0063 meters
Now, it is mentioned that the distance between a node and an adjacent antinode represents half a wavelength (n = 0.5).
Substituting the known values into the formula, we have:
λ = 2 * 0.0063 / 0.5 = 0.0252 meters
The wavelength (λ) is given, and we can use the formula:
c = f * λ
Where c is the speed of light and f is the frequency.
The speed of light (c) is approximately 3.00 x 10^8 meters per second.
Substituting the known values into the formula, we have:
3.00 x 10^8 = f * 0.0252
Solving for f:
f = (3.00 x 10^8) / 0.0252
f ≈ 1.19 x 10^10 Hz (rounded to two significant figures)
Therefore, the microwave frequency is approximately 1.19 x 10^10 Hz.
To determine the microwave frequency in this case, we can start by understanding the concept of standing waves. In a standing wave pattern, certain points along the wave remain stationary, called nodes, while other points undergo maximum displacement, called antinodes. The distance between a node and an adjacent antinode is referred to as half a wavelength.
Given that the distance between a node and an adjacent antinode in this case is 0.63 cm, we know that it corresponds to half a wavelength. Therefore, the full wavelength would be twice this distance, or 2 × 0.63 cm.
Now, we need to convert the wavelength from centimeters to meters before calculating the frequency. Since the speed of light in a vacuum is approximately 3.00 × 10^8 meters per second, we can use the formula:
speed of light = frequency × wavelength
Let's convert the distance first:
2 × 0.63 cm = 1.26 cm = 1.26 × 10^(-2) m
Now, let's calculate the frequency:
speed of light = frequency × wavelength
3.00 × 10^8 m/s = frequency × 1.26 × 10^(-2) m
Rearranging the formula to solve for frequency:
frequency = (speed of light) / (wavelength)
Plugging in the values:
frequency = (3.00 × 10^8 m/s) / (1.26 × 10^(-2) m)
Calculating this, we find:
frequency = 2.38 × 10^10 Hz
Therefore, the microwave frequency in this standing wave pattern is approximately 2.38 × 10^10 Hz.