Problem 56 - Physics

Question

(a) Suppose that the magnitude \(E\) of the electric field in an electromagnetic wave triples. By what factor does the intensity \(S\) of the wave change? (b) The magnitude \(B\) of the magnetic field is much smaller than \(E\) because, according to Equation 24.3, \(B=E / c\), where \(c\) is the speed of light in a vacuum. If \(B\) triples, by what factor does the intensity change? Account for your answers.

Step-by-Step Solution

Answer

Both scenarios lead to the intensity increasing by a factor of 9 when the field triples.

Step 1: Understand the Relationship Between Intensity and Electric Field

The intensity of an electromagnetic wave, denoted as \( S \), is related to the square of the electric field magnitude \( E \) by the formula \( S = \frac{1}{2} c \varepsilon_0 E^2 \), where \( c \) is the speed of light and \( \varepsilon_0 \) is the permittivity of free space. Thus, if the electric field triples, the intensity becomes \( S' = \frac{1}{2} c \varepsilon_0 (3E)^2 = 9 \times \frac{1}{2} c \varepsilon_0 E^2 = 9S \). Hence, the intensity changes by a factor of 9.

Step 2: Analyze the Relationship Between Magnetic Field and Intensity

The magnetic field \( B \) of an electromagnetic wave is related to the electric field by the equation \( B = \frac{E}{c} \). Substituting this into our intensity formula, we see that \( S = \frac{1}{2} c \varepsilon_0 (cB)^2 = \frac{1}{2} c^3 \varepsilon_0 B^2 \). If \( B \) triples, then \( B' = 3B \), so the intensity becomes \( S' = \frac{1}{2} c^3 \varepsilon_0 (3B)^2 = 9 \times \frac{1}{2} c^3 \varepsilon_0 B^2 = 9S \). Hence, the intensity changes by a factor of 9.

Step 3: Conclusion

In both scenarios (electric field and magnetic field tripling), the intensity of the electromagnetic wave increases by a factor of 9 because intensity is proportional to the square of each field's magnitude.

Key Concepts

Electric Field Intensity

In an electromagnetic wave, the electric field intensity plays a crucial role. It is represented by the magnitude \( E \), and it determines many characteristics of the wave. Electromagnetic waves, such as light and radio waves, consist of electric and magnetic fields vibrating at right angles to each other. The electric field component influences how the wave interacts with materials and travels through different mediums.

The intensity \( S \), or the wave's power per unit area, is directly related to the electric field. Specifically, the intensity is proportional to the square of the electric field magnitude. This relationship is given by the formula:

• \( S = \frac{1}{2} c \varepsilon_0 E^2 \)

where:

• \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{m/s} \)
• \( \varepsilon_0 \) is the permittivity of free space, a constant value \( 8.85 \times 10^{-12} \text{F/m} \)
• \( E \) is the electric field magnitude

If the magnitude \( E \) of the electric field triples, then the intensity \( S \) changes by a factor of 9, due to the \( E^2 \) relationship.

Magnetic Field Relation

The magnetic field, noted as \( B \), is inherently linked to the electric field in an electromagnetic wave. This connection is described by a simple equation: \( B = \frac{E}{c} \). Here, \( B \) is much smaller in magnitude than \( E \) because of the factor \( \frac{1}{c} \), where \( c \) is the speed of light.

In terms of intensity, the energy carried by both fields contributes to the wave's complete description. By substituting \( E = cB \) into the intensity formula, a clear understanding emerges:

• \( S = \frac{1}{2} c^3 \varepsilon_0 B^2 \)

where the intensity now also includes \( B \). If the magnetic field magnitude \( B \) triples, similar to the electric field, the intensity \( S \) increases ninefold, reinforcing the squared dependency in the formula. This symmetry between \( E \) and \( B \)'s squared relationship highlights the beauty of electromagnetic wave properties.

Intensity Proportionality

Intensity proportionality is a fundamental aspect of electromagnetic waves. It links the wave's measurable energy to the square of either the electric or magnetic field's magnitude. This quadratic relationship shows that any change in the field magnitude results in a significant change in intensity.

For a clearer perspective, here’s how it works:

• If the electric field \( E \) changes, the intensity changes by \( E^2 \)
• If the magnetic field \( B \) changes, the intensity changes by \( B^2 \)
• Any proportional changes in either field will cause the intensity to alter by the square of that factor

This concept of intensity proportionality is critical in understanding waves and forms the basis of many optical and electromagnetic phenomena. It emphasizes the interdependence between field magnitude and the resulting wave energy.

Electromagnetic Theory

Electromagnetic theory forms the foundation of understanding how electric and magnetic fields interact to produce waves. Described initially in Maxwell's equations, this theory explains a wide array of physical phenomena, from the behavior of light to radio signals.

Key points in electromagnetic theory include:

• Electromagnetic waves are transverse, with electric and magnetic fields oscillating perpendicular to wave direction
• Energy is equally distributed between the electric and magnetic fields in a wave
• The speed of electromagnetic waves in a vacuum is constant and represented by \( c \)

Understanding electromagnetic theory helps clarify why electromagnetic waves change intensity when either the electric or magnetic field magnitude changes. It provides insight into our everyday interactions with electromagnetic waves, explaining how technologies like wireless communication and optics function.