The wave length (λ) in meters of an electromagnetic wave is related to its frequency (f) in MHz as:

Question

The wave length (&lambda;) in meters of an electromagnetic wave is related to its frequency (f) in MHz as:

Prepp Team · Accepted Answer

Wavelength Calculation for Electromagnetic Waves

Understanding the relationship between wavelength, frequency, and the speed of an electromagnetic wave is fundamental in physics and electronics. Electromagnetic waves, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, all travel at the speed of light in a vacuum.

Electromagnetic Wave Basics

The speed of any wave is related to its wavelength and frequency by a simple equation. For electromagnetic waves traveling in a vacuum, this speed is the speed of light, often denoted by $c$.

Speed of light ($c$): In a vacuum, the speed of light is approximately $3 \times 10^8$ meters per second (m/s). This is a universal constant.
Wavelength ($\lambda$): This is the spatial period of the wave, the distance over which the wave's shape repeats. It is typically measured in meters (m).
Frequency ($f$): This is the number of wave cycles that pass a point per unit time. It is typically measured in Hertz (Hz), where 1 Hz means one cycle per second.

Formula Relating Wavelength, Frequency, and Speed of Light

The fundamental relationship between these three quantities is given by:

$c = \lambda f$

Where:

$c$ is the speed of light.
$\lambda$ is the wavelength.
$f$ is the frequency.

From this formula, we can express the wavelength $\lambda$ as:

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

Units Conversion for Frequency

The question provides the frequency ($f$) in Megahertz (MHz). To use the speed of light in meters per second and obtain the wavelength in meters, we must convert the frequency from MHz to Hertz (Hz).

1 Megahertz (MHz) = $10^6$ Hertz (Hz)

So, if the frequency is $f$ in MHz, its value in Hz will be $f \times 10^6$ Hz.

Step-by-Step Wavelength Calculation

Let's substitute the values and units into the formula:

Start with the fundamental formula:
$\lambda = \frac{c}{f}$
Substitute the value of the speed of light:
$c = 3 \times 10^8$ m/s
$\lambda = \frac{3 \times 10^8 \text{ m/s}}{f \text{ (in Hz)}}$
Convert frequency from MHz to Hz:
Let $f_{MHz}$ be the frequency in MHz. Then $f_{Hz} = f_{MHz} \times 10^6$ Hz.
$\lambda = \frac{3 \times 10^8 \text{ m/s}}{(f_{MHz} \times 10^6) \text{ Hz}}$
Simplify the expression:
$\lambda = \frac{3 \times 10^8}{f_{MHz} \times 10^6}$
Perform the division of powers of 10:
$\lambda = \frac{3 \times 10^{(8-6)}}{f_{MHz}}$
Final simplified formula:
$\lambda = \frac{3 \times 10^2}{f_{MHz}}$
$\lambda = \frac{300}{f_{MHz}}$

Therefore, the wavelength $\lambda$ in meters of an electromagnetic wave is related to its frequency $f$ in MHz as $\lambda = \frac{300}{f}$.

Comparing with Options

Let's compare our derived formula with the given options:

Option	Formula	Analysis
1	$\lambda = \frac{3\times10^8}{f}$	Incorrect. This formula would be correct if $f$ were in Hz, not MHz.
2	$\lambda = \frac{3\times10^{10}}{f}$	Incorrect. The power of 10 is too high; it suggests a speed of $3 \times 10^{10}$ m/s, or an incorrect unit conversion factor.
3	$\lambda = \frac{300}{f}$	Correct. This matches our derived formula after converting frequency from MHz to Hz.
4	None of the above	Incorrect, as option 3 is correct.

The correct relationship is $\lambda = \frac{300}{f}$, where $\lambda$ is in meters and $f$ is in MHz.

Prepp Team · Answer

$\lambda = \frac{3\times10^8}{f}$

Prepp Team · Answer

$\lambda = \frac{3\times10^{10}}{f}$

Prepp Team · Answer

None of the above