A microwave beam with a wavelength of 1.5 cm has an intensity of 25 W/m2. What is the magnetic field amplitude?

Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

문제 21

31장, 문제 21

A microwave beam with a wavelength of 1.5 cm has an intensity of 25 W/m². What is the magnetic field amplitude?

검증된 단계별 안내

Step 1: Recall the relationship between the intensity of an electromagnetic wave and the amplitudes of the electric and magnetic fields. The intensity (I) is given by the formula:

I = \(\frac{1}{2}\) \(\epsilon\)_0 c E_0^2

, where

\(\epsilon\)_0

is the permittivity of free space,

c

is the speed of light, and

E_0

is the electric field amplitude.

Step 2: Use the relationship between the electric field amplitude (

E_0

) and the magnetic field amplitude (

B_0

) in an electromagnetic wave:

E_0 = c B_0

. Rearrange this equation to express

B_0

in terms of

E_0

B_0 = \(\frac{E_0}{c}\)

Step 3: Solve for the electric field amplitude (

E_0

) using the intensity formula. Rearrange the formula to isolate

E_0

E_0 = \(\sqrt{\frac{2I}{\epsilon_0 c}\)}

. Substitute the given intensity (

I = 25 \ \(\text{W/m}\)^2

), the permittivity of free space (

\(\epsilon\)_0 = 8.85 \(\times\) 10^{-12} \ \(\text{F/m}\)

), and the speed of light (

c = 3 \(\times\) 10^8 \ \(\text{m/s}\)

) into this equation.

Step 4: Once

E_0

is calculated, substitute it into the equation

B_0 = \(\frac{E_0}{c}\)

to find the magnetic field amplitude. Use the value of

c = 3 \(\times\) 10^8 \ \(\text{m/s}\)

for the speed of light.

Step 5: Perform the calculations step by step to determine the numerical value of

B_0

. Ensure that all units are consistent throughout the calculations.

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주요 개념

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Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. They consist of a transverse wave where the electric field (E) and magnetic field (B) oscillate perpendicular to each other and the direction of wave propagation. Understanding the nature of these waves is crucial for analyzing phenomena like microwaves, which are a type of electromagnetic radiation.

Intensity of Electromagnetic Waves

The intensity of an electromagnetic wave is defined as the power per unit area carried by the wave, typically measured in watts per square meter (W/m²). It is related to the electric and magnetic field amplitudes through the equation I = (1/2) * ε₀ * c * E², where ε₀ is the permittivity of free space and c is the speed of light. This relationship is essential for calculating the magnetic field amplitude from the given intensity.

Magnetic Field Amplitude

The magnetic field amplitude (B₀) of an electromagnetic wave can be derived from the electric field amplitude (E₀) using the relationship B₀ = E₀/c, where c is the speed of light. This relationship highlights how the electric and magnetic fields are interconnected in electromagnetic waves. Knowing the intensity allows us to find E₀, and subsequently B₀, which is necessary for solving the problem presented.