9.2 Poynting vector

The Poynting vector is a crucial concept in electromagnetism, representing the directional energy flow in electromagnetic fields. It quantifies the rate and direction of energy transfer per unit area, providing insights into power transmission and wave propagation.

Derived from Maxwell's equations, the Poynting vector is the cross product of electric and magnetic fields. It plays a key role in understanding energy conservation, wave intensity, and practical applications like waveguides and electromagnetic shielding.

Definition of Poynting vector

• The Poynting vector, denoted as $\vec{S}$, represents the directional energy flux density of an electromagnetic field
• Quantifies the rate and direction of electromagnetic energy flow per unit area perpendicular to the direction of propagation
• Introduced by John Henry Poynting in 1884 to describe the energy transport in electromagnetic fields

Derivation from Maxwell's equations

• The Poynting vector can be derived from Maxwell's equations by considering the work done by electromagnetic fields
• Starting with Faraday's law and Ampère's law, the derivation involves taking the cross product of the electric field $\vec{E}$ and magnetic field $\vec{H}$
• The resulting expression for the Poynting vector is $\vec{S} = \vec{E} \times \vec{H}$, where $\vec{E}$ is the electric field and $\vec{H}$ is the magnetic field intensity

Units and dimensions

• The SI unit of the Poynting vector is watts per square meter (W/m²)
• Dimensionally, the Poynting vector is expressed as $[S] = [E][H] = [M][L]^2[T]^{-3}[I]^{-1}$, where $[M]$, $[L]$, $[T]$, and $[I]$ represent mass, length, time, and electric current, respectively
• The magnitude of the Poynting vector gives the instantaneous power density at a given point in space

Direction of energy flow

• The direction of the Poynting vector indicates the direction of energy flow in an electromagnetic field
• Energy propagates in the direction perpendicular to both the electric and magnetic field vectors
• The right-hand rule can be used to determine the direction of the Poynting vector based on the orientation of the electric and magnetic fields

Relationship to electric and magnetic fields

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• The Poynting vector is always perpendicular to both the electric and magnetic field vectors at any point in space
• The magnitude of the Poynting vector is proportional to the product of the magnitudes of the electric and magnetic fields
• In electromagnetic waves, the electric and magnetic fields are mutually perpendicular, and the Poynting vector is perpendicular to both, indicating the direction of wave propagation

Poynting vector in electromagnetic waves

• The Poynting vector plays a crucial role in describing energy flow in electromagnetic waves
• In electromagnetic waves, the Poynting vector is always in the direction of wave propagation
• The magnitude of the Poynting vector in electromagnetic waves represents the intensity of the wave, which is the power per unit area

Plane waves

• For plane electromagnetic waves, the Poynting vector is constant in magnitude and direction at any point in space
• In a plane wave, the electric and magnetic fields are in phase and perpendicular to each other and the direction of propagation
• The magnitude of the Poynting vector in a plane wave is given by $S = \frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}E_0^2$, where $\varepsilon$ is the permittivity, $\mu$ is the permeability, and $E_0$ is the amplitude of the electric field

Spherical waves

• For spherical electromagnetic waves, the Poynting vector points radially outward from the source
• The magnitude of the Poynting vector in spherical waves decreases with the square of the distance from the source
• In the far-field region, spherical waves can be approximated as plane waves, and the Poynting vector becomes perpendicular to the wavefront

Time-averaged Poynting vector

• The time-averaged Poynting vector, denoted as $\langle\vec{S}\rangle$, represents the average power density over one period of the electromagnetic wave
• For time-harmonic fields, the time-averaged Poynting vector is given by $\langle\vec{S}\rangle = \frac{1}{2}\text{Re}(\vec{E} \times \vec{H}^*)$, where $\vec{H}^*$ is the complex conjugate of the magnetic field intensity
• The time-averaged Poynting vector is particularly useful for analyzing the power flow in steady-state conditions

Relation to intensity

• The magnitude of the time-averaged Poynting vector is equal to the intensity of the electromagnetic wave
• Intensity is a measure of the average power per unit area carried by the wave
• For plane waves, the intensity is given by $I = \frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}E_0^2$, which is the same as the magnitude of the time-averaged Poynting vector

Energy conservation and Poynting's theorem

• Poynting's theorem is a statement of energy conservation in electromagnetic fields
• It relates the Poynting vector to the rate of change of electromagnetic energy density and the work done by the fields
• Poynting's theorem can be expressed in both integral and differential forms

Integral form

• The integral form of Poynting's theorem states that the net power flow through a closed surface is equal to the negative rate of change of the electromagnetic energy stored within the volume enclosed by the surface minus the work done by the fields on the charges within the volume
• Mathematically, it is expressed as $\oint_S \vec{S} \cdot d\vec{A} = -\frac{d}{dt} \int_V (u_E + u_H) dV - \int_V \vec{J} \cdot \vec{E} dV$, where $u_E$ and $u_H$ are the electric and magnetic energy densities, respectively, and $\vec{J}$ is the current density

Differential form

• The differential form of Poynting's theorem is obtained by applying the divergence theorem to the integral form
• It relates the divergence of the Poynting vector to the rate of change of the electromagnetic energy density and the work done by the fields at each point in space
• The differential form is given by $\nabla \cdot \vec{S} = -\frac{\partial u}{\partial t} - \vec{J} \cdot \vec{E}$, where $u = u_E + u_H$ is the total electromagnetic energy density

Applications of Poynting vector

• The Poynting vector has numerous applications in various fields of electromagnetics and optics
• It is used to analyze power flow, energy transfer, and the interaction of electromagnetic fields with matter
• Some notable applications include power transmission in waveguides, radiation pressure, and electromagnetic shielding

Power transmission in waveguides

• The Poynting vector is used to calculate the power flow in waveguides, such as rectangular or circular waveguides
• By integrating the Poynting vector over the cross-sectional area of the waveguide, the total power transmitted can be determined
• This information is crucial for designing efficient waveguide systems and optimizing power delivery

Radiation pressure and solar sails

• The Poynting vector is related to the concept of radiation pressure, which is the pressure exerted by electromagnetic waves on a surface
• Radiation pressure is proportional to the magnitude of the Poynting vector and can be used to calculate the force exerted by light on objects
• Solar sails, which are spacecraft propelled by radiation pressure from sunlight, rely on the principles of the Poynting vector to generate thrust

Electromagnetic shielding effectiveness

• The Poynting vector is used to evaluate the effectiveness of electromagnetic shielding materials and enclosures
• By calculating the ratio of the Poynting vector magnitudes on the outside and inside of the shield, the shielding effectiveness can be determined
• This information is essential for designing protective enclosures and minimizing electromagnetic interference in sensitive electronic devices

Poynting vector vs energy density

• The Poynting vector and electromagnetic energy density are related but distinct concepts
• The Poynting vector represents the rate and direction of energy flow per unit area, while energy density quantifies the amount of energy stored per unit volume in the electromagnetic field

Differences and similarities

• The Poynting vector is a vector quantity, whereas energy density is a scalar quantity
• Both the Poynting vector and energy density are functions of the electric and magnetic field strengths
• The Poynting vector is related to the rate of change of energy density through Poynting's theorem
• In the absence of energy dissipation or work done by the fields, the divergence of the Poynting vector is equal to the negative rate of change of the energy density

Poynting vector in dispersive media

• In dispersive media, where the permittivity and permeability are frequency-dependent, the Poynting vector expression needs to be modified
• The dispersive nature of the medium affects the propagation of electromagnetic waves and the energy flow
• The modified Poynting vector in dispersive media takes into account the frequency-dependent material properties

Modifications for frequency-dependent permittivity and permeability

• In dispersive media, the permittivity $\varepsilon(\omega)$ and permeability $\mu(\omega)$ are functions of the angular frequency $\omega$
• The modified Poynting vector is given by $\vec{S} = \frac{1}{2}\text{Re}(\vec{E} \times \vec{H}^*) + \frac{1}{2}\omega\frac{\partial(\varepsilon'\mu')}{\partial\omega}|\vec{E}|^2\hat{k}$, where $\varepsilon'$ and $\mu'$ are the real parts of the permittivity and permeability, respectively, and $\hat{k}$ is the unit vector in the direction of wave propagation
• The additional term in the modified Poynting vector accounts for the energy stored in the medium due to its dispersive properties

Experimental measurement techniques

• Measuring the Poynting vector experimentally is crucial for verifying theoretical predictions and understanding energy flow in practical scenarios
• Several techniques have been developed to measure the Poynting vector, including near-field scanning optical microscopy (NSOM) and electromagnetic field probes

Near-field scanning optical microscopy (NSOM)

• NSOM is a technique that allows for high-resolution mapping of the Poynting vector in the near-field region of electromagnetic fields
• It uses a small aperture or a sharp probe tip to collect light from the surface of the sample, overcoming the diffraction limit of conventional optical microscopy
• By scanning the probe over the sample surface and measuring the collected light, a high-resolution map of the Poynting vector can be obtained

Electromagnetic field probes

• Electromagnetic field probes are devices designed to measure the electric and magnetic field components at a specific point in space
• These probes can be based on various principles, such as small antennas, Hall effect sensors, or electro-optic crystals
• By measuring the electric and magnetic fields simultaneously, the Poynting vector can be calculated using the cross product of the field vectors
• Field probes offer a direct way to measure the Poynting vector in real-time and can be used in a wide range of frequencies and field strengths