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118-500: In Poynting's original paper and in most textbooks, the Poynting vector S {\displaystyle \mathbf {S} } is defined as the cross product S = E × H , {\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} ,} where bold letters represent vectors and This expression is often called the Abraham form and

236-416: A ‖ ‖ b ‖ | sin ⁡ θ | . {\displaystyle \left\|\mathbf {a} \times \mathbf {b} \right\|=\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\left|\sin \theta \right|.} Indeed, one can also compute the volume V of a parallelepiped having a , b and c as edges by using a combination of a cross product and

354-406: A k 2 2 η . {\displaystyle \left\langle {\mathsf {S_{z}}}\right\rangle ={\frac {\mathsf {E_{peak}^{2}}}{2\eta }}.} This is the most common form for the energy flux of a plane wave, since sinusoidal field amplitudes are most often expressed in terms of their peak values, and complicated problems are typically solved considering only one frequency at

472-401: A and b is defined only in three-dimensional space and is denoted by a × b . In physics and applied mathematics , the wedge notation a ∧ b is often used (in conjunction with the name vector product ), although in pure mathematics such notation is usually reserved for just the exterior product, an abstraction of the vector product to n dimensions. The cross product a × b

590-403: A and b is the zero vector 0 . The direction of the vector n depends on the chosen orientation of the space. Conventionally, it is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b . Then, the vector n is coming out of the thumb (see the adjacent picture). Using this rule implies that

708-420: A and b . Conversely, a dot product a ⋅ b involves multiplications between corresponding components of a and b . As explained below , the cross product can be expressed in the form of a determinant of a special 3 × 3 matrix. According to Sarrus's rule , this involves multiplications between matrix elements identified by crossed diagonals. If ( i , j , k ) is a positively oriented orthonormal basis,

826-449: A displacement current ; therefore it stores and returns electrical energy as if it were an ideal capacitor. The electric susceptibility χ e {\displaystyle \chi _{e}} of a dielectric material is a measure of how easily it polarises in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from

944-477: A pseudovector . See § Handedness for more detail. In 1842, William Rowan Hamilton first described the algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to

1062-467: A ∥ b ) so that the sine of the angle between them is zero ( θ = 0° or θ = 180° and sin  θ = 0 ). The self cross product of a vector is the zero vector: The cross product is anticommutative , distributive over addition, and compatible with scalar multiplication so that It is not associative , but satisfies the Jacobi identity : Distributivity, linearity and Jacobi identity show that

1180-546: A completely generalized view. The Poynting vector appears in Poynting's theorem (see that article for the derivation), an energy-conservation law: ∂ u ∂ t = − ∇ ⋅ S − J f ⋅ E , {\displaystyle {\frac {\partial u}{\partial t}}=-\mathbf {\nabla } \cdot \mathbf {S} -\mathbf {J_{\mathrm {f} }} \cdot \mathbf {E} ,} where J f

1298-408: A conductor. The definition and computation of the speed of light in a conductor can be given. Inside the conductor, the Poynting vector represents energy flow from the electromagnetic field into the wire, producing resistive Joule heating in the wire. For a derivation that starts with Snell's law see Reitz page 454. The density of the linear momentum of the electromagnetic field is S / c where S

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1416-437: A consequence of causality , imposes Kramers–Kronig constraints on the real and imaginary parts of the susceptibility χ e ( ω ) {\displaystyle \chi _{e}(\omega )} . In the classical approach to the dielectric, the material is made up of atoms. Each atom consists of a cloud of negative charge (electrons) bound to and surrounding a positive point charge at its center. In

1534-400: A dot product, called scalar triple product (see Figure 2): Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value: Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of perpendicularity in the same way that the dot product is

1652-465: A frequency-dependent response of a medium for wave propagation. When the frequency becomes higher: In the frequency region above ultraviolet, permittivity approaches the constant ε 0 in every substance, where ε 0 is the permittivity of the free space. Because permittivity indicates the strength of the relation between an electric field and polarisation, if a polarisation process loses its response, permittivity decreases. Dielectric relaxation

1770-456: A full cycle T = 2 π / ω . The following quantity, still referred to as a "Poynting vector", is expressed directly in terms of the phasors as: S m = 1 2 E m × H m ∗ , {\displaystyle \mathbf {S} _{\mathrm {m} }={\tfrac {1}{2}}\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*},} where denotes

1888-549: A material cannot polarise instantaneously in response to an applied field. The more general formulation as a function of time is P ( t ) = ε 0 ∫ − ∞ t χ e ( t − t ′ ) E ( t ′ ) d t ′ . {\displaystyle \mathbf {P} (t)=\varepsilon _{0}\int _{-\infty }^{t}\chi _{e}\left(t-t'\right)\mathbf {E} (t')\,dt'.} That is,

2006-416: A measure of parallelism . Given two unit vectors , their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel. Unit vectors enable two convenient identities: the dot product of two unit vectors yields

2124-805: A net current of zero (+ I in the center conductor and − I in the outer conductor), and again the electric field is zero there anyway. Using Ampère's law in the region from R 1 to R 2 , which encloses the current + I in the center conductor but with no contribution from the current in the outer conductor, we find at radius r : I = ∮ C H ⋅ d s = 2 π r H θ ( r ) H θ ( r ) = I 2 π r {\displaystyle {\begin{aligned}I=\oint _{C}\mathbf {H} \cdot ds&=2\pi rH_{\theta }(r)\\H_{\theta }(r)&={\frac {I}{2\pi r}}\end{aligned}}} Now, from an electric field in

2242-408: A result, when lattice vibrations or molecular vibrations induce relative displacements of the atoms, the centers of positive and negative charges are also displaced. The locations of these centers are affected by the symmetry of the displacements. When the centers do not correspond, polarisation arises in molecules or crystals. This polarisation is called ionic polarisation . Ionic polarisation causes

2360-408: A scale factor t , leading to: for some scalar t . If, in addition to a × b = a × c and a ≠ 0 as above, it is the case that a ⋅ b = a ⋅ c then As b − c cannot be simultaneously parallel (for the cross product to be 0 ) and perpendicular (for the dot product to be 0) to a , it must be the case that b and c cancel: b = c . From the geometrical definition,

2478-478: A section of coaxial cable analyzed in cylindrical coordinates as depicted in the accompanying diagram. We can take advantage of the model's symmetry: no dependence on θ (circular symmetry) nor on Z (position along the cable). The model (and solution) can be considered simply as a DC circuit with no time dependence, but the following solution applies equally well to the transmission of radio frequency power, as long as we are considering an instant of time (during which

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2596-448: A specified dielectric constant ε r , or in optics with a material whose refractive index n = ϵ r {\displaystyle {\mathsf {n}}={\sqrt {\epsilon _{r}}}} , the intrinsic impedance is found as: η = η 0 ϵ r . {\displaystyle \eta ={\frac {\eta _{0}}{\sqrt {\epsilon _{r}}}}.} In optics,

2714-3036: A standing wave which can be described as two such waves travelling in opposite directions), E and H are exactly in phase, so S m is simply a real number according to the above definition. The equivalence of Re( S m ) to the time-average of the instantaneous Poynting vector S can be shown as follows. S ( t ) = E ( t ) × H ( t ) = Re ( E m e j ω t ) × Re ( H m e j ω t ) = 1 2 ( E m e j ω t + E m ∗ e − j ω t ) × 1 2 ( H m e j ω t + H m ∗ e − j ω t ) = 1 4 ( E m × H m ∗ + E m ∗ × H m + E m × H m e 2 j ω t + E m ∗ × H m ∗ e − 2 j ω t ) = 1 2 Re ( E m × H m ∗ ) + 1 2 Re ( E m × H m e 2 j ω t ) . {\displaystyle {\begin{aligned}\mathbf {S} (t)&=\mathbf {E} (t)\times \mathbf {H} (t)\\&=\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }e^{j\omega t}\right)\times \operatorname {Re} \!\left(\mathbf {H} _{\mathrm {m} }e^{j\omega t}\right)\\&={\tfrac {1}{2}}\!\left(\mathbf {E} _{\mathrm {m} }e^{j\omega t}+\mathbf {E} _{\mathrm {m} }^{*}e^{-j\omega t}\right)\times {\tfrac {1}{2}}\!\left(\mathbf {H} _{\mathrm {m} }e^{j\omega t}+\mathbf {H} _{\mathrm {m} }^{*}e^{-j\omega t}\right)\\&={\tfrac {1}{4}}\!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*}+\mathbf {E} _{\mathrm {m} }^{*}\times \mathbf {H} _{\mathrm {m} }+\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }e^{2j\omega t}+\mathbf {E} _{\mathrm {m} }^{*}\times \mathbf {H} _{\mathrm {m} }^{*}e^{-2j\omega t}\right)\\&={\tfrac {1}{2}}\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*}\right)+{\tfrac {1}{2}}\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }e^{2j\omega t}\right)\!.\end{aligned}}} The average of

2832-551: A three-dimensional oriented Euclidean vector space (named here E {\displaystyle E} ), and is denoted by the symbol × {\displaystyle \times } . Given two linearly independent vectors a and b , the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b , and thus normal to the plane containing them. It has many applications in mathematics, physics , engineering , and computer programming . It should not be confused with

2950-411: A time. However, the expression using E rms is totally general, applying, for instance, in the case of noise whose RMS amplitude can be measured but where the "peak" amplitude is meaningless. In free space the intrinsic impedance η is simply given by the impedance of free space η 0 ≈   377   Ω. In non-magnetic dielectrics (such as all transparent materials at optical frequencies) with

3068-487: A vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. The cross-product in seven dimensions has undesirable properties (e.g. it fails to satisfy the Jacobi identity ), so it is not used in mathematical physics to represent quantities such as multi-dimensional space-time . (See § Generalizations below for other dimensions.) The cross product of two vectors

3186-734: Is ∂ u ∂ t = − ∇ ⋅ S − J ⋅ E , {\displaystyle {\frac {\partial u}{\partial t}}=-\nabla \cdot \mathbf {S} -\mathbf {J} \cdot \mathbf {E} ,} where J is the total current density and the energy density u is given by u = 1 2 ( ε 0 | E | 2 + 1 μ 0 | B | 2 ) , {\displaystyle u={\frac {1}{2}}\!\left(\varepsilon _{0}|\mathbf {E} |^{2}+{\frac {1}{\mu _{0}}}|\mathbf {B} |^{2}\right)\!,} where ε 0

3304-720: Is ⟨ S ⟩ c 2 {\displaystyle {\frac {\langle S\rangle }{c^{2}}}} For linear, nondispersive and isotropic (for simplicity) materials, the constitutive relations can be written as D = ε E , B = μ H , {\displaystyle \mathbf {D} =\varepsilon \mathbf {E} ,\quad \mathbf {B} =\mu \mathbf {H} ,} where Here ε and μ are scalar, real-valued constants independent of position, direction, and frequency. In principle, this limits Poynting's theorem in this form to fields in vacuum and nondispersive linear materials. A generalization to dispersive materials

3422-511: Is not acceptable to add an arbitrary solenoidal field to E × H . The consideration of the Poynting vector in static fields shows the relativistic nature of the Maxwell equations and allows a better understanding of the magnetic component of the Lorentz force , q ( v × B ) . To illustrate, the accompanying picture is considered, which describes the Poynting vector in a cylindrical capacitor, which

3540-405: Is a vector , William Kingdon Clifford coined the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature. Both the cross notation ( a × b ) and the name cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of

3658-453: Is a delay or lag in the response of a linear system , and therefore dielectric relaxation is measured relative to the expected linear steady state (equilibrium) dielectric values. The time lag between electrical field and polarisation implies an irreversible degradation of Gibbs free energy . In physics , dielectric relaxation refers to the relaxation response of a dielectric medium to an external, oscillating electric field. This relaxation

Poynting vector - Misplaced Pages Continue

3776-433: Is applied, the distance between charges within each permanent dipole, which is related to chemical bonding , remains constant in orientation polarisation; however, the direction of polarisation itself rotates. This rotation occurs on a timescale that depends on the torque and surrounding local viscosity of the molecules. Because the rotation is not instantaneous, dipolar polarisations lose the response to electric fields at

3894-415: Is defined as a vector c that is perpendicular (orthogonal) to both a and b , with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. The cross product is defined by the formula where If the vectors a and b are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of

4012-401: Is equal to its power divided by c , the speed of light. Therefore, the circular flow of electromagnetic energy implies an angular momentum. If one were to connect a wire between the two plates of the charged capacitor, then there would be a Lorentz force on that wire while the capacitor is discharging due to the discharge current and the crossed magnetic field; that force would be tangential to

4130-986: Is given above in terms of the center conductor voltage V . The total power flowing down the coaxial cable can be computed by integrating over the entire cross section A of the cable in between the conductors: P tot = ∬ A S z ( r , θ ) d A = ∫ R 2 R 1 2 π r d r S z ( r ) = ∫ R 2 R 1 W I r d r = W I ln ⁡ ( R 2 R 1 ) . {\displaystyle {\begin{aligned}P_{\text{tot}}&=\iint _{\mathbf {A} }S_{z}(r,\theta )\,dA=\int _{R_{2}}^{R_{1}}2\pi rdrS_{z}(r)\\&=\int _{R_{2}}^{R_{1}}{\frac {W\,I}{r}}dr=W\,I\,\ln \left({\frac {R_{2}}{R_{1}}}\right).\end{aligned}}} Substituting

4248-458: Is located in an H field (pointing into the page) generated by a permanent magnet. Although there are only static electric and magnetic fields, the calculation of the Poynting vector produces a clockwise circular flow of electromagnetic energy, with no beginning or end. While the circulating energy flow may seem unphysical, its existence is necessary to maintain conservation of angular momentum . The momentum of an electromagnetic wave in free space

4366-551: Is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem , the integral becomes a simple product, P ( ω ) = ε 0 χ e ( ω ) E ( ω ) . {\displaystyle \mathbf {P} (\omega )=\varepsilon _{0}\chi _{e}(\omega )\mathbf {E} (\omega ).} The susceptibility (or equivalently

4484-498: Is not gained from or lost to other forms of energy within some region (e.g., mechanical energy, or heat), then electromagnetic energy is locally conserved within that region, yielding a continuity equation as a special case of Poynting's theorem: ∇ ⋅ S = − ∂ u ∂ t {\displaystyle \nabla \cdot \mathbf {S} =-{\frac {\partial u}{\partial t}}} where u {\displaystyle u}

4602-454: Is often described in terms of permittivity as a function of frequency , which can, for ideal systems, be described by the Debye equation. On the other hand, the distortion related to ionic and electronic polarisation shows behaviour of the resonance or oscillator type. The character of the distortion process depends on the structure, composition, and surroundings of the sample. Debye relaxation

4720-422: Is only required that the surface integral of the Poynting vector around a closed surface describe the net flow of electromagnetic energy into or out of the enclosed volume. This means that adding a solenoidal vector field (one with zero divergence) to S will result in another field that satisfies this required property of a Poynting vector field according to Poynting's theorem. Since the divergence of any curl

4838-404: Is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor , because they have no loosely bound, or free, electrons that may drift through the material, but instead they shift, only slightly, from their average equilibrium positions, causing dielectric polarisation . Because of dielectric polarisation , positive charges are displaced in

Poynting vector - Misplaced Pages Continue

4956-420: Is possible under certain circumstances at the cost of additional terms. One consequence of the Poynting formula is that for the electromagnetic field to do work, both magnetic and electric fields must be present. The magnetic field alone or the electric field alone cannot do any work. In a propagating electromagnetic plane wave in an isotropic lossless medium, the instantaneous Poynting vector always points in

5074-611: Is related to the polarisation density P {\displaystyle \mathbf {P} } by D   =   ε 0 E + P   =   ε 0 ( 1 + χ e ) E   =   ε 0 ε r E . {\displaystyle \mathbf {D} \ =\ \varepsilon _{0}\mathbf {E} +\mathbf {P} \ =\ \varepsilon _{0}\left(1+\chi _{e}\right)\mathbf {E} \ =\ \varepsilon _{0}\varepsilon _{r}\mathbf {E} .} In general,

5192-456: Is roughly the inverse of the time it takes for the molecules to bend, and this distortion polarisation disappears above the infrared. Ionic polarisation is polarisation caused by relative displacements between positive and negative ions in ionic crystals (for example, NaCl ). If a crystal or molecule consists of atoms of more than one kind, the distribution of charges around an atom in the crystal or molecule leans to positive or negative. As

5310-418: Is the current density of free charges and u is the electromagnetic energy density for linear, nondispersive materials, given by u = 1 2 ( E ⋅ D + B ⋅ H ) , {\displaystyle u={\frac {1}{2}}\!\left(\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} \right)\!,} where The first term in

5428-590: Is the electric permittivity of free space . The susceptibility of a medium is related to its relative permittivity ε r {\displaystyle \varepsilon _{r}} by χ e   = ε r − 1. {\displaystyle \chi _{e}\ =\varepsilon _{r}-1.} So in the case of a classical vacuum , χ e   = 0. {\displaystyle \chi _{e}\ =0.} The electric displacement D {\displaystyle \mathbf {D} }

5546-457: Is the root mean square (RMS) electric field amplitude. In the important case that E ( t ) is sinusoidally varying at some frequency with peak amplitude E peak , E rms is E p e a k / 2 {\displaystyle {\mathsf {E_{peak}}}/{\sqrt {2}}} , with the average Poynting vector then given by: ⟨ S z ⟩ = E p e

5664-483: Is the transpose of the inverse and cof {\displaystyle \operatorname {cof} } is the cofactor matrix. It can be readily seen how this formula reduces to the former one if M {\displaystyle M} is a rotation matrix. If M {\displaystyle M} is a 3-by-3 symmetric matrix applied to a generic cross product a × b {\displaystyle \mathbf {a} \times \mathbf {b} } ,

5782-417: Is the vacuum permittivity . It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only. The two alternative definitions of the Poynting vector are equal in vacuum or in non-magnetic materials, where B = μ 0 H . In all other cases, they differ in that S = (1/ μ 0 ) E × B and the corresponding u are purely radiative, since

5900-603: Is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity ε of a medium as a function of the field's angular frequency ω : ε ^ ( ω ) = ε ∞ + Δ ε 1 + i ω τ , {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{1+i\omega \tau }},} where ε ∞

6018-584: Is the double-frequency component having an average value of zero, so we find: ⟨ S ⟩ = Re ( 1 2 E m × H m ∗ ) = Re ( S m ) {\displaystyle \langle \mathbf {S} \rangle =\operatorname {Re} \!\left({\tfrac {1}{2}}{\mathbf {E} _{\mathrm {m} }}\times \mathbf {H} _{\mathrm {m} }^{*}\right)=\operatorname {Re} \!\left(\mathbf {S} _{\mathrm {m} }\right)} According to some conventions,

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6136-526: Is the electrically insulating material between the metallic plates of a capacitor . The polarisation of the dielectric by the applied electric field increases the capacitor's surface charge for the given electric field strength. The term dielectric was coined by William Whewell (from dia + electric ) in response to a request from Michael Faraday . A perfect dielectric is a material with zero electrical conductivity ( cf. perfect conductor infinite electrical conductivity), thus exhibiting only

6254-427: Is the energy density of the electromagnetic field. This frequent condition holds in the following simple example in which the Poynting vector is calculated and seen to be consistent with the usual computation of power in an electric circuit. Although problems in electromagnetics with arbitrary geometries are notoriously difficult to solve, we can find a relatively simple solution in the case of power transmission through

6372-481: Is the magnitude of the Poynting vector and c is the speed of light in free space. The radiation pressure exerted by an electromagnetic wave on the surface of a target is given by P r a d = ⟨ S ⟩ c . {\displaystyle P_{\mathrm {rad} }={\frac {\langle S\rangle }{\mathrm {c} }}.} The Poynting vector occurs in Poynting's theorem only through its divergence ∇ ⋅ S , that is, it

6490-468: Is the momentary delay (or lag) in the dielectric constant of a material. This is usually caused by the delay in molecular polarisation with respect to a changing electric field in a dielectric medium (e.g., inside capacitors or between two large conducting surfaces). Dielectric relaxation in changing electric fields could be considered analogous to hysteresis in changing magnetic fields (e.g., in inductor or transformer cores ). Relaxation in general

6608-404: Is the most widely used. The Poynting vector is usually denoted by S or N . In simple terms, the Poynting vector S depicts the direction and rate of transfer of energy, that is power , due to electromagnetic fields in a region of space that may or may not be empty. More rigorously, it is the quantity that must be used to make Poynting's theorem valid. Poynting's theorem essentially says that

6726-419: Is the permittivity at the high frequency limit, Δ ε = ε s − ε ∞ where ε s is the static, low frequency permittivity, and τ is the characteristic relaxation time of the medium. Separating into the real part ε ′ {\displaystyle \varepsilon '} and the imaginary part ε ″ {\displaystyle \varepsilon ''} of

6844-420: Is zero , one can add the curl of any vector field to the Poynting vector and the resulting vector field S ′ will still satisfy Poynting's theorem. However even though the Poynting vector was originally formulated only for the sake of Poynting's theorem in which only its divergence appears, it turns out that the above choice of its form is unique. The following section gives an example which illustrates why it

6962-409: Is zero. The cross product is anticommutative (that is, a × b = − b × a ) and is distributive over addition, that is, a × ( b + c ) = a × b + a × c . The space E {\displaystyle E} together with the cross product is an algebra over the real numbers , which is neither commutative nor associative , but is a Lie algebra with the cross product being

7080-449: The Lie bracket . Like the dot product, it depends on the metric of Euclidean space , but unlike the dot product, it also depends on a choice of orientation (or " handedness ") of the space (it is why an oriented space is needed). The resultant vector is invariant of rotation of basis. Due to the dependence on handedness , the cross product is said to be a pseudovector . In connection with

7198-501: The R vector space together with vector addition and the cross product forms a Lie algebra , the Lie algebra of the real orthogonal group in 3 dimensions, SO(3) . The cross product does not obey the cancellation law ; that is, a × b = a × c with a ≠ 0 does not imply b = c , but only that: This can be the case where b and c cancel, but additionally where a and b − c are parallel; that is, they are related by

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7316-477: The dot product (projection product). The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or if either one has zero length, then their cross product

7434-465: The ferroelectric effect as well as dipolar polarisation. The ferroelectric transition, which is caused by the lining up of the orientations of permanent dipoles along a particular direction, is called an order-disorder phase transition . The transition caused by ionic polarisations in crystals is called a displacive phase transition . Ionic polarisation enables the production of energy-rich compounds in cells (the proton pump in mitochondria ) and, at

7552-533: The plasma membrane , the establishment of the resting potential , energetically unfavourable transport of ions, and cell-to-cell communication (the Na+/K+-ATPase ). All cells in animal body tissues are electrically polarised – in other words, they maintain a voltage difference across the cell's plasma membrane , known as the membrane potential . This electrical polarisation results from a complex interplay between ion transporters and ion channels . In neurons,

7670-790: The vector norm of A . Since E and H are at right angles to each other, the magnitude of their cross product is the product of their magnitudes. Without loss of generality let us take X to be the direction of the electric field and Y to be the direction of the magnetic field. The instantaneous Poynting vector, given by the cross product of E and H will then be in the positive Z direction: | S z | = | E x H y | = | E x | 2 η . {\displaystyle \left|{\mathsf {S_{z}}}\right|=\left|{\mathsf {E_{x}}}{\mathsf {H_{y}}}\right|={\frac {\left|{\mathsf {E_{x}}}\right|^{2}}{\eta }}.} Finding

7788-428: The θ direction, that is, a vector field looping around the center conductor at every radius between R 1 and R 2 . Inside the conductors themselves the magnetic field may or may not be zero, but this is of no concern since the Poynting vector in these regions is zero due to the electric field's being zero. Outside the entire coaxial cable, the magnetic field is identically zero since paths in this region enclose

7906-462: The Poynting vector represents the instantaneous power flow due to instantaneous electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms of sinusoidally varying fields at a specified frequency. The results can then be applied more generally, for instance, by representing incoherent radiation as a superposition of such waves at different frequencies and with fluctuating amplitudes. We would thus not be considering

8024-407: The Poynting vector, however, we are able to identify the profile of power flow in terms of the electric and magnetic fields inside the coaxial cable. The electric fields are of course zero inside of each conductor, but in between the conductors ( R 1 < r < R 2 {\displaystyle R_{1}<r<R_{2}} ) symmetry dictates that they are strictly in

8142-750: The Poynting vector, or use D and H to construct yet another version. The choice has been controversial: Pfeifer et al. summarize and to a certain extent resolve the century-long dispute between proponents of the Abraham and Minkowski forms (see Abraham–Minkowski controversy ). The Poynting vector represents the particular case of an energy flux vector for electromagnetic energy. However, any type of energy has its direction of movement in space, as well as its density, so energy flux vectors can be defined for other types of energy as well, e.g., for mechanical energy . The Umov–Poynting vector discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in

8260-400: The above equation for ε ^ ( ω ) {\displaystyle {\hat {\varepsilon }}(\omega )} is sometimes written with 1 − i ω τ {\displaystyle 1-i\omega \tau } in the denominator due to an ongoing sign convention ambiguity whereby many sources represent the time dependence of

8378-481: The above-mentioned equalities and collecting similar terms, we obtain: meaning that the three scalar components of the resulting vector s = s 1 i + s 2 j + s 3 k = a × b are Using column vectors , we can represent the same result as follows: The cross product can also be expressed as the formal determinant: This determinant can be computed using Sarrus's rule or cofactor expansion . Using Sarrus's rule, it expands to which gives

8496-415: The basis vectors satisfy the following equalities which imply, by the anticommutativity of the cross product, that The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that These equalities, together with the distributivity and linearity of the cross product (though neither follows easily from the definition given above), are sufficient to determine

8614-472: The cable contains an ideal dielectric material of relative permittivity ε r and we assume conductors that are non-magnetic (so μ = μ 0 ) and lossless (perfect conductors), all of which are good approximations to real-world coaxial cable in typical situations. The center conductor is held at voltage V and draws a current I toward the right, so we expect a total power flow of P = V · I according to basic laws of electricity . By evaluating

8732-594: The capacitance of capacitors to the speed of light . It is defined as the constant of proportionality (which may be a tensor ) relating an electric field E {\displaystyle \mathbf {E} } to the induced dielectric polarisation density P {\displaystyle \mathbf {P} } such that P = ε 0 χ e E , {\displaystyle \mathbf {P} =\varepsilon _{0}\chi _{e}\mathbf {E} ,} where ε 0 {\displaystyle \varepsilon _{0}}

8850-416: The central axis and thus add angular momentum to the system. That angular momentum would match the "hidden" angular momentum, revealed by the Poynting vector, circulating before the capacitor was discharged. Cross product In mathematics , the cross product or vector product (occasionally directed area product , to emphasize its geometric significance) is a binary operation on two vectors in

8968-407: The complex conjugate. The time-averaged power flow (according to the instantaneous Poynting vector averaged over a full cycle, for instance) is then given by the real part of S m . The imaginary part is usually ignored, however, it signifies "reactive power" such as the interference due to a standing wave or the near field of an antenna. In a single electromagnetic plane wave (rather than

9086-842: The complex dielectric permittivity yields: ε ′ = ε ∞ + ε s − ε ∞ 1 + ω 2 τ 2 ε ″ = ( ε s − ε ∞ ) ω τ 1 + ω 2 τ 2 {\displaystyle {\begin{aligned}\varepsilon '&=\varepsilon _{\infty }+{\frac {\varepsilon _{s}-\varepsilon _{\infty }}{1+\omega ^{2}\tau ^{2}}}\\[3pt]\varepsilon ''&={\frac {(\varepsilon _{s}-\varepsilon _{\infty })\omega \tau }{1+\omega ^{2}\tau ^{2}}}\end{aligned}}} Note that

9204-792: The complex electric field with exp ⁡ ( − i ω t ) {\displaystyle \exp(-i\omega t)} whereas others use exp ⁡ ( + i ω t ) {\displaystyle \exp(+i\omega t)} . In the former convention, the functions ε ′ {\displaystyle \varepsilon '} and ε ″ {\displaystyle \varepsilon ''} representing real and imaginary parts are given by ε ^ ( ω ) = ε ′ + i ε ″ {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon '+i\varepsilon ''} whereas in

9322-398: The components of the resulting vector directly. The latter formula avoids having to change the orientation of the space when we inverse an orthonormal basis. The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides (see Figure 1): ‖ a × b ‖ = ‖

9440-432: The correct average power flow is obtained without multiplication by 1/2. If a conductor has significant resistance, then, near the surface of that conductor, the Poynting vector would be tilted toward and impinge upon the conductor. Once the Poynting vector enters the conductor, it is bent to a direction that is almost perpendicular to the surface. This is a consequence of Snell's law and the very slow speed of light inside

9558-415: The cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive). If the cross product of two vectors is the zero vector (that is, a × b = 0 ), then either one or both of the inputs is the zero vector, ( a = 0 or b = 0 ) or else they are parallel or antiparallel (

9676-409: The cross product is anti-commutative ; that is, b × a = −( a × b ) . By pointing the forefinger toward b first, and then pointing the middle finger toward a , the thumb will be forced in the opposite direction, reversing the sign of the product vector. As the cross product operator depends on the orientation of the space, in general the cross product of two vectors is not a "true" vector, but

9794-411: The cross product is invariant under proper rotations about the axis defined by a × b . In formulae: More generally, the cross product obeys the following identity under matrix transformations: where M {\displaystyle M} is a 3-by-3 matrix and ( M − 1 ) T {\displaystyle \left(M^{-1}\right)^{\mathrm {T} }}

9912-562: The cross product of any two vectors a and b . Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors: Their cross product a × b can be expanded using distributivity: This can be interpreted as the decomposition of a × b into the sum of nine simpler cross products involving vectors aligned with i , j , or k . Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using

10030-406: The cross product, the exterior product of vectors can be used in arbitrary dimensions (with a bivector or 2-form result) and is independent of the orientation of the space. The product can be generalized in various ways, using the orientation and metric structure just as for the traditional 3-dimensional cross product; one can, in n dimensions, take the product of n − 1 vectors to produce

10148-528: The dielectric now depends on the situation. The more complicated the situation, the richer the model must be to accurately describe the behaviour. Important questions are: The relationship between the electric field E and the dipole moment M gives rise to the behaviour of the dielectric, which, for a given material, can be characterised by the function F defined by the equation: M = F ( E ) . {\displaystyle \mathbf {M} =\mathbf {F} (\mathbf {E} ).} When both

10266-423: The dielectric. (Note that the dipole moment points in the same direction as the electric field in the figure. This is not always the case, and is a major simplification, but is true for many materials.) When the electric field is removed, the atom returns to its original state. The time required to do so is called relaxation time; an exponential decay. This is the essence of the model in physics. The behaviour of

10384-448: The difference between the electromagnetic energy entering a region and the electromagnetic energy leaving a region must equal the energy converted or dissipated in that region, that is, turned into a different form of energy (often heat). So if one accepts the validity of the Poynting vector description of electromagnetic energy transfer, then Poynting's theorem is simply a statement of the conservation of energy . If electromagnetic energy

10502-505: The direction of propagation while rapidly oscillating in magnitude. This can be simply seen given that in a plane wave, the magnitude of the magnetic field H ( r , t ) is given by the magnitude of the electric field vector E ( r , t ) divided by η , the intrinsic impedance of the transmission medium: | H | = | E | η , {\displaystyle |\mathbf {H} |={\frac {|\mathbf {E} |}{\eta }},} where | A | represents

10620-630: The direction of the field and negative charges shift in the direction opposite to the field. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarised, but also reorient so that their symmetry axes align to the field. The study of dielectric properties concerns storage and dissipation of electric and magnetic energy in materials. Dielectrics are important for explaining various phenomena in electronics , optics , solid-state physics and cell biophysics . Although

10738-500: The dissipation term − J ⋅ E covers the total current, while the E × H definition has contributions from bound currents which are then excluded from the dissipation term. Since only the microscopic fields E and B occur in the derivation of S = (1/ μ 0 ) E × B and the energy density, assumptions about any material present are avoided. The Poynting vector and theorem and expression for energy density are universally valid in vacuum and all materials. The above form for

10856-418: The earlier solution for the constant W we find: P t o t = I ln ⁡ ( R 2 R 1 ) V ln ⁡ ( R 2 / R 1 ) = V I {\displaystyle P_{\mathrm {tot} }=I\ln \left({\frac {R_{2}}{R_{1}}}\right){\frac {V}{\ln(R_{2}/R_{1})}}=V\,I} that is,

10974-446: The factor of 1/2 in the above definition may be left out. Multiplication by 1/2 is required to properly describe the power flow since the magnitudes of E m and H m refer to the peak fields of the oscillating quantities. If rather the fields are described in terms of their root mean square (RMS) values (which are each smaller by the factor 2 / 2 {\displaystyle {\sqrt {2}}/2} ), then

11092-407: The following relation holds true: The cross product of two vectors lies in the null space of the 2 × 3 matrix with the vectors as rows: For the sum of two cross products, the following identity holds: Dielectric In electromagnetism , a dielectric (or dielectric medium ) is an electrical insulator that can be polarised by an applied electric field . When a dielectric material

11210-436: The frequency of an applied electric field. Because there is a lag between changes in polarisation and changes in the electric field, the permittivity of the dielectric is a complex function of the frequency of the electric field. Dielectric dispersion is very important for the applications of dielectric materials and the analysis of polarisation systems. This is one instance of a general phenomenon known as material dispersion :

11328-445: The highest frequencies. A molecule rotates about 1 radian per picosecond in a fluid, thus this loss occurs at about 10 Hz (in the microwave region). The delay of the response to the change of the electric field causes friction and heat. When an external electric field is applied at infrared frequencies or less, the molecules are bent and stretched by the field and the molecular dipole moment changes. The molecular vibration frequency

11446-447: The instantaneous E ( t ) and H ( t ) used above, but rather a complex (vector) amplitude for each which describes a coherent wave's phase (as well as amplitude) using phasor notation. These complex amplitude vectors are not functions of time, as they are understood to refer to oscillations over all time. A phasor such as E m is understood to signify a sinusoidally varying field whose instantaneous amplitude E ( t ) follows

11564-1001: The instantaneous Poynting vector S over time is given by: ⟨ S ⟩ = 1 T ∫ 0 T S ( t ) d t = 1 T ∫ 0 T [ 1 2 Re ( E m × H m ∗ ) + 1 2 Re ( E m × H m e 2 j ω t ) ] d t . {\displaystyle \langle \mathbf {S} \rangle ={\frac {1}{T}}\int _{0}^{T}\mathbf {S} (t)\,dt={\frac {1}{T}}\int _{0}^{T}\!\left[{\tfrac {1}{2}}\operatorname {Re} \!\left(\mathbf {E} _{\mathrm {m} }\times \mathbf {H} _{\mathrm {m} }^{*}\right)+{\tfrac {1}{2}}\operatorname {Re} \!\left({\mathbf {E} _{\mathrm {m} }}\times {\mathbf {H} _{\mathrm {m} }}e^{2j\omega t}\right)\right]dt.} The second term

11682-999: The latter convention ε ^ ( ω ) = ε ′ − i ε ″ {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon '-i\varepsilon ''} . The above equation uses the latter convention. The dielectric loss is also represented by the loss tangent: tan ⁡ ( δ ) = ε ″ ε ′ = ( ε s − ε ∞ ) ω τ ε s + ε ∞ ω 2 τ 2 {\displaystyle \tan(\delta )={\frac {\varepsilon ''}{\varepsilon '}}={\frac {\left(\varepsilon _{s}-\varepsilon _{\infty }\right)\omega \tau }{\varepsilon _{s}+\varepsilon _{\infty }\omega ^{2}\tau ^{2}}}} This relaxation model

11800-414: The negative of dot product and cross product of the two vectors. In 1881, Josiah Willard Gibbs , and independently Oliver Heaviside , introduced the notation for both the dot product and the cross product using a period ( a ⋅ b ) and an "×" ( a × b ), respectively, to denote them. In 1877, to emphasize the fact that the result of a dot product is a scalar while the result of a cross product

11918-610: The negative of the voltage V : − V = ∫ R 2 R 1 W r d r = − W ln ⁡ ( R 2 R 1 ) {\displaystyle -V=\int _{R_{2}}^{R_{1}}{\frac {W}{r}}dr=-W\ln \left({\frac {R_{2}}{R_{1}}}\right)} so that: W = V ln ⁡ ( R 2 / R 1 ) {\displaystyle W={\frac {V}{\ln(R_{2}/R_{1})}}} The magnetic field, again by symmetry, can only be non-zero in

12036-405: The nuclei is possible (distortion polarisation). Orientation polarisation results from a permanent dipole, e.g., that arises from the 104.45° angle between the asymmetric bonds between oxygen and hydrogen atoms in the water molecule, which retains polarisation in the absence of an external electric field. The assembly of these dipoles forms a macroscopic polarisation. When an external electric field

12154-476: The permittivity) is frequency dependent. The change of susceptibility with respect to frequency characterises the dispersion properties of the material. Moreover, the fact that the polarisation can only depend on the electric field at previous times (i.e., χ e ( Δ t ) = 0 {\displaystyle \chi _{e}(\Delta t)=0} for Δ t < 0 {\displaystyle \Delta t<0} ),

12272-829: The polarisation is a convolution of the electric field at previous times with time-dependent susceptibility given by χ e ( Δ t ) {\displaystyle \chi _{e}(\Delta t)} . The upper limit of this integral can be extended to infinity as well if one defines χ e ( Δ t ) = 0 {\displaystyle \chi _{e}(\Delta t)=0} for Δ t < 0 {\displaystyle \Delta t<0} . An instantaneous response corresponds to Dirac delta function susceptibility χ e ( Δ t ) = χ e δ ( Δ t ) {\displaystyle \chi _{e}(\Delta t)=\chi _{e}\delta (\Delta t)} . It

12390-401: The power given by integrating the Poynting vector over a cross section of the coaxial cable is exactly equal to the product of voltage and current as one would have computed for the power delivered using basic laws of electricity. Other similar examples in which the P = V · I result can be analytically calculated are: the parallel-plate transmission line, using Cartesian coordinates , and

12508-413: The presence of an electric field, the charge cloud is distorted, as shown in the top right of the figure. This can be reduced to a simple dipole using the superposition principle . A dipole is characterised by its dipole moment , a vector quantity shown in the figure as the blue arrow labeled M . It is the relationship between the electric field and the dipole moment that gives rise to the behaviour of

12626-435: The radial direction and it can be shown (using Gauss's law ) that they must obey the following form: E r ( r ) = W r {\displaystyle E_{r}(r)={\frac {W}{r}}} W can be evaluated by integrating the electric field from r = R 2 {\displaystyle r=R_{2}} to R 1 {\displaystyle R_{1}} which must be

12744-658: The radial direction, and a tangential magnetic field, the Poynting vector, given by the cross-product of these, is only non-zero in the Z direction, along the direction of the coaxial cable itself, as we would expect. Again only a function of r , we can evaluate S (r): S z ( r ) = E r ( r ) H θ ( r ) = W r I 2 π r = W I 2 π r 2 {\displaystyle S_{z}(r)=E_{r}(r)H_{\theta }(r)={\frac {W}{r}}{\frac {I}{2\pi r}}={\frac {W\,I}{2\pi r^{2}}}} where W

12862-399: The real part of E m   e where ω is the (radian) frequency of the sinusoidal wave being considered. In the time domain, it will be seen that the instantaneous power flow will be fluctuating at a frequency of 2 ω . But what is normally of interest is the average power flow in which those fluctuations are not considered. In the math below, this is accomplished by integrating over

12980-418: The right-hand side represents the electromagnetic energy flow into a small volume, while the second term subtracts the work done by the field on free electrical currents, which thereby exits from electromagnetic energy as dissipation , heat, etc. In this definition, bound electrical currents are not included in this term and instead contribute to S and u . For light in free space, the linear momentum density

13098-411: The term insulator implies low electrical conduction , dielectric typically means materials with a high polarisability . The latter is expressed by a number called the relative permittivity . Insulator is generally used to indicate electrical obstruction while dielectric is used to indicate the energy storing capacity of the material (by means of polarisation). A common example of a dielectric

13216-573: The time-averaged power in the plane wave then requires averaging over the wave period (the inverse frequency of the wave): ⟨ S z ⟩ = ⟨ | E x | 2 ⟩ η = E rms 2 η , {\displaystyle \left\langle {\mathsf {S_{z}}}\right\rangle ={\frac {\left\langle \left|{\mathsf {E_{x}}}\right|^{2}\right\rangle }{\eta }}={\frac {\mathsf {E_{\text{rms}}^{2}}}{\eta }},} where E rms

13334-463: The two-wire transmission line, using bipolar cylindrical coordinates . In the "microscopic" version of Maxwell's equations, this definition must be replaced by a definition in terms of the electric field E and the magnetic flux density B (described later in the article). It is also possible to combine the electric displacement field D with the magnetic flux B to get the Minkowski form of

13452-405: The type of electric field and the type of material have been defined, one then chooses the simplest function F that correctly predicts the phenomena of interest. Examples of phenomena that can be so modelled include: Dipolar polarisation is a polarisation that is either inherent to polar molecules (orientation polarisation), or can be induced in any molecule in which the asymmetric distortion of

13570-411: The types of ion channels in the membrane usually vary across different parts of the cell, giving the dendrites , axon , and cell body different electrical properties. As a result, some parts of the membrane of a neuron may be excitable (capable of generating action potentials), whereas others are not. In physics, dielectric dispersion is the dependence of the permittivity of a dielectric material on

13688-449: The vacuum permittivity and permeability are used, and there is no D or H . When this model is used, the Poynting vector is defined as S = 1 μ 0 E × B , {\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,} where This is actually the general expression of the Poynting vector. The corresponding form of Poynting's theorem

13806-415: The value of radiated flux crossing a surface, thus the average Poynting vector component in the direction normal to that surface, is technically known as the irradiance , more often simply referred to as the intensity (a somewhat ambiguous term). The "microscopic" (differential) version of Maxwell's equations admits only the fundamental fields E and B , without a built-in model of material media. Only

13924-410: The voltage and current don't change), and over a sufficiently short segment of cable (much smaller than a wavelength, so that these quantities are not dependent on Z ). The coaxial cable is specified as having an inner conductor of radius R 1 and an outer conductor whose inner radius is R 2 (its thickness beyond R 2 doesn't affect the following analysis). In between R 1 and R 2

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