Chapter 7 Time-varying Electromagnetic Fields

Chapter 7 Time-varying Electromagnetic Fields Displacement Current, Maxwell’s Equations Boundary Conditions, Potential Function Energy Flow Density, Time-harmonic Electromagnetic Fields Complex Vector Expressions 1. Displacement Electric Current 2. Maxwell’s Equations 3. Boundary Conditions for Time-varying Electromagnetic Fields 4. Scalar and Vector Potentials 5. Solution of Equations for Potentials 6. Energy Density and Energy Flow Density Vector 7. Uniqueness Theorem 8. Time-harmonic Electromagnetic Fields 9. Complex Maxwell’s Equations 10. Complex Potentials 11. Complex Energy Density and Energy Flow Density Vector

For static fields 1. Displacement Electric Current The displacement current is neither the conduction current nor the convection current, which are formed by the motion of electric charges.It is a concept given by J. C. Maxwell. Based on the principle of electric charge conservation, we have which are called the continuity equations for electric current.

Gauss’ law for electrostatic fields, , is still valid for time-varying electric fields, we obtain  Obviously, the dimension of is the same as that of the current density. For time-varying electromagnetic fields, since the charges are changing with time, the electric current continuity principle cannot be derived from static considerations. Nevertheless, an electric current is always continuous. Hence an extension of earliest concepts for steady current need to be developed. The current in a vacuum capacitor is neither the conduction current nor the convection current, but it is actually the displacement electric current.

British scientist, James Clerk Maxwell named the density of the displacement current, denoted as Jd, so that We obtain For electrostatic fields, , and the displacement current is zero. In imperfect dielectrics, , while in a good conductor, . The introduction of the displacement current makes the time-varying total current continuous, and the above equations are called the principle of total current continuity. The density of the displacement current is the time rate of change of the electric flux density, hence In time-varying electric fields, the displacement current is larger if the electric field is changing more rapidly.

i.e. Maxwell considered that the displacement current must also produce magnetic fields, and it should be included in the Ampere circuital law, so that Which are Ampere’s circuital law with the displacement current. It shows that a time-varying magnetic field is produced by the conduction current, the convection current, and the displacement current. The displacement current, which results from time-varying electric field, produces a time-varying magnetic field. The law of electromagnetic induction shows that a time-varying magnetic field can produce a time-varying electric field. Maxwell deduced the coexistence of a time-varying electric field and a time-varying magnetic field, and they result in an electromagneticwave in space. This prediction was demonstrated in 1888 by Hertz.

The integral form The differential form 2. Maxwell’s Equations For the time-varying electromagnetic field,Maxwell summarized the following four equations:

The time-varying electric field is both divergent and curly, and the time-varying magnetic field is solenoidal and curly. Nevertheless, the time-varying electric field and the time-varying magnetic field cannot be separated, and the time-varying electromagnetic field is divergent and curly. In a source-free region, the time-varying electromagnetic field is solenoidal. The electric field lines and the magnetic field lines are linked with each other, forming closed loops, and resulting in an electromagnetic wave in space. The time-varying electric field and the time-varying magnetic field are perpendicular to each other.

where stands for the impressed source producing the time-varying electromagnetic field. In order to describe more completely the behavior of time-varying electromagnetic fields, Maxwell’s equations need to be supplemented by the charge conservation equation and the constitutive relations: The four Maxwell’s equations are not independent. Equations 4 and 3 can be derived from Equation 1 and 2, respectively, and vise versa. For static fields, we have Maxwell’s equations become the former equations for electrostatic field and steady magnetic field. Furthermore, the electric field and the magnetic field are independent each other.

As the founder of relativity, Albert Einstein (1879-1955), pointed out in his book “The Evolution of Physics” that “The formulation of these equations is the most important event in physics since Newton’s time, and they are the quantitative mathematical description of the laws of the field. Their content is much richer than we have been able to indicate, and the simple form conceals a depth revealed only by careful study”. “These equations are the laws representing the structure of the field. They do not, as in Newton’s laws, connect two widely separated events; they do not connect the happenings here with the conditions there”. “The field here and now depends on the field in the immediate neighborhood at a time just past. The equations allow us to predict what will happen a little further in space and a little later in time, if we know what happens here and now”

Maxwell’s equations have made important impact on the history of mankind, besides the advancement of science and technology. As American physicist, Richard P. Feynman, said in his book “The Feynman Lectures on Physics”, that “From along view of the history of mankind──seen from say, ten thousand years from now ──there can be little doubt that the most significant event of the 19-th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American civil war will pale into provincial insignificance in comparison with this important scientific event of the same decade.”

In this information age, from baby monitor to remote-controlled equipment, from radar to microwave oven, from radio broadcast to satellite TV, from ground mobile communication to space communi-cation, from wireless local area network to blue tooth technology, and from global positioning to navigation systems, electromagnetic waves are used as to make these technologies possible. The wireless information highway makes it possible that we can reach anybody anywhere at anytime, and to be able to send text, voice, or video signals to the recipient miles away. Electromagnetic waves can recreate the experience of events far away, making possible wireless virtual reality. To appreciate the contributions Maxwell and Hertz have made to the progress of mankind and our culture, one only needs to look at the wide usage of electromagnetic waves.

en ② or ① 3. Boundary Conditions for Time-varying Electromagnetic Fields In principle, all boundary conditions satisfied by a static field can be applied to a time-varying electromagnetic field. (a)The tangential components of the electric field intensity are continuous at any boundary, i.e. As long as the time rate of change of the magnetic flux density is finite, using the same method as before we can obtain it from the equation: For linear isotropic media, the above equation can be rewritten as

or For linear isotropic media, we have or (b) The normal components of magnetic flux intensity are continuous at any boundary. From the principle of magnetic flux continuity,we find (c) The boundary condition for the normal components of electric flux density depends on the property of the media. In general, from Gauss’ law we find where S is the surface density of the free charge at the boundary.

For a linear isotropic dielectric, we have or At the boundary between two perfect dielectrics, because of the absence of free charges, we have (d) The boundary condition for the tangential components of the magnetic field intensity depends also on the property of the media. In general, in the absence of surface currents at the boundary, as long as the time rate of change of the electric flux density is finite, we find However, surface currents can exist on the surface of a perfect electric conductor, and in this case the tangential components of the magnetic field intensity are discontinuous.

Assume the boundary is formed by a perfect dielectric and a perfect electric conductor.Time-varying electromagnetic field and conduction current cannot exist in the perfect electric conductor, and they are able to be only on the surface of the perfect electric conductor.  J=E   E ≠ 0     H ≠ 0 E ≠ 0 E(t), B (t), J (t)= 0  H ≠ 0 J ≠ 0 The tangential component of the electric field intensity and the normal component of magnetic field intensity are continuous at any boundaries, and they cannot exist on the surface of a perfect electric conductor. Consequently, the time-varying electric field must be perpendicular to the surface of the perfect electric conductor, while the time-varying magnetic field is tangential to the surface.

en E ② H2t JS et  , H ① H1t  Due to ，we find or Since there exists surface currentJS on the surface of perfect electric conductor,considering the direction of the surface current density and the integral path complying with the right hand rule, , we find or

            y z    a  y g x x z b b a y z x Electric field lines Magnetic field lines Example. The components of the time-varying electromagnetic field in a rectangular metal waveguide of interior cross-sectiona  b are Find the density of the displacement current in the waveguide and the charges and the currents on the interior walls of the waveguide. The inside is vacuum.

Solution:(a) We find the displacement current as (b) At the interior wall y = 0，we have At the interior wall y = b, we have

At the lateral wall x = 0 , , we have At the lateral wall x = a , , we find At the lateral walls x = 0 and x = a, due to , and . y The currents on the interior walls z x

Using , and considering and ，then the above equations become 4. Scalar and Vector Potentials Suppose the medium is linear, homogeneous, and isotropic, from Maxwell’s equation we find

Due to , hence B can be expressed in terms of the curl of a vector field A, as given by where Ais called the vector potential. Substituting the above equation into gives The relationship between the field intensities and the sources is quite complicated. To simplify the process, it will be helpful to solve the time-varying electromagnetic fields by introducing two auxiliary functions: the scalar and the vector potentials.

The above equation can be rewritten as The vector is irrotational. Thus it can be expressed in terms of the gradient of a scalar , so that where is called the scalar potential, and we have The vector potential Aand the scalar potential are functions of time and space. If they are independent of time, then the results are the same as that of the static fields. Therefore, the vector potential Ais also called the vector magnetic potential and the scalar potential is also called the scalar electric potential.。

Using , the above equations become In order to derive the relationship between the potentials and the sources, from the definition of the potentials and Maxwell’s equations we obtain

The curl of the vector field A is given as , but the divergence must be specified. In principle, the divergence can be taken arbitrarily, but to simplify the application of the equations, we can see that if let Lorentz condition Then the above two equations become After the divergence of the vector potential A is given by the Lorentz condition, the equations are simplified. The original equations are two coupled equations, while new equations are decoupled. The vector potential Aonly depends on the currentJ, while the scalar potential is related to the charge density only.

If the current and the charge are known, then the vector potential A and the scalar potential  can be determined. After A and are found, the electric and the magnetic fields can be obtained. Consequently, the solution of Maxwell’s equations is related to that of the equations for the potential functions, and the solution is simplified. The original Equations are two vector equations with complicated structure, and in tree-dimensional space, six coordinate components need to be solved. New potential equations are a vector equation and a scalar equation, respectively. In three-dimensional space, only four coordinate components need to be found. In particular, in rectangular coordinate system the vector equation can be resolved into three scalar equations。

where 5. Solution of Equations for Potentials Here we find the solution by using an analogous method based on the results of static fields. The scalar potential caused by a point charge is obtained first, then use superposition principle to obtain the solution of the scalar potential due to a distribution of time-varying charge. If the source is a time-varying point charge placed at the origin, the distribution of the field should be a function of the variable ronly, and independent of the angles  and  . In the open space excluding the origin, the scalar potential function satisfies the following equation

The electric potential produced by the static elemental charge at the origin is The above equation is the homogeneous wave equation for the function ( r), and the general solution is We will know that the second term is contrary to the physical situation, and it should be excluded. Therefore, we find the scalar electric potential as Comparing the above two equations, we know

z  (r, t) r' - r dV' V ' r' r O y x Hence, we find the electric potential produced by the time-varying elemental charge at the origin as where r is the distance to the field point from the charge dV . From the above result, the electric potential produced by the volume charge in V  can be obtained as

To find the vector potential function A, the above equation can be expanded in rectangular coordinate system, with all coordinate components satisfying the same inhomogeneous wave equation, i.e. Apparently, for each component we can find a solution similar to that of scalar potential equation. Incorporating the three components gives the solution of the vector potential A as

Both equations show that the solution of the scalar or the vector potential at the momentt is related to the source distribution at the moment . In other words, the field at t does not depend on the source at the same moment, but on the source at an earlier time . It means that the field produced by the source at rneeds a certain time to reach r, and this time difference is . The quantity is the distance between the source point and the field point, and vstands for the propagation velocity of the electromagnetic wave.

From we can see that the propagation velocity of electromagnetic wave is related to the properties of the medium. In vacuum, which is the propagation velocity of light in vacuum, also called the speed of light, usually denoted as c. It is worth noting that the field at a point away from the source may still be present at a moment after the source ceases to exist. Energy released by a source travels away from the source and continuous to propagate even after the source is taken away. This phenomenon is a consequence of electromagnetic radiation.

Since the time factor implies that the evolution of the field precedes that of the source, it violates causality, and must be abandoned. The time factor can be rewritten as Hence, the function can be considered as a wave traveling toward the origin as a reflected wave from a distant location. The change with respect to time in the scalar electric potential and the vector magnetic potential A is always lagging behind the sources. Hence the functions  andA are called the retarded potentials. For a point charge placed in open free space this reflective wave cannot exist.

We obtain For surface or line charge and current, consider the following substitutions: All of the above equations are valid only for linear, homogeneous, and isotropic media.

All the formulas for the density of electrostatic energy , the density of steady magnetic energy , and the density of power dissipation pl for a steady current field are valid for time-varying electromagnetic fields. 6. Energy Density and Energy Flow Density Vector For a linear isotropic media The density of energy of a time-varying electromagnetic field is given by Since a time-varying electromagnetic field changes with respect to space and time, the density of its energy is also a function of space and time, giving rise to a flow of energy.。

Suppose there is a linear isotropic medium in the region Vwithout any impressed source , and the parameters of the medium are . Then in this region the electromagnetic field satisfies the following Maxwell’s equations: , ,  V E, H Using , we have Integrating both sides of the equation over the volume V, we have

Considering , we have The vector ( ) represents the power passing through unit cross-sectional area and that is just the energy flow density vector S, given by S , ,  E, H Based on the definition of energy density, the above equation can be rewritten as which is called the energy theorem for time-varying electromagnetic field, and any time-varying electromagnetic field satisfying Maxwell’s equation must obey the energy theorem.

This equation states that and . S, E, H are perpendicular to each other in space. They comply with the right hand rule. E S H The instantaneous value of the energy flow density is The instantaneous value of the energy flow density is equal to the product of the instantaneous electric field intensity and the instantaneous magnetic field intensity. The energy flow density is maximum only if the two field intensities are maximum. If the electric field intensity or the magnetic field intensity is zero at a certain moment, the energy flow density will be zero at that moment.

S E t (r, t) or H t (r, t) V 7. Uniqueness Theorem In a region Vbound by a closed surface S, if the initial values of the electric field intensity E and the magnetic field intensity H are given at the time t = 0, as long as the tangential component Et of the electric field intensity or the tangential componentHtof the magnetic field intensity is given at the boundary for t > 0, then the electromagnetic field will be uniquely determined by Maxwell’s equations at any moment t (t > 0) in the region V. E( r, t), H(r, t ) E(r, 0) & H(r, 0 ) Based on the energy theorem and using the method of contrary, this theorem can be proved.

8. Time-harmonic Electromagnetic Fields A special kind of time-varying electromagnetic field that has sinusoidal time dependence, with its general mathematical form given by where Em(r) is a function of space only, referred to as the amplitude of the sinusoidal function. is the angular frequency and e(r)is the initial phase. The sinusoidal electromagnetic field is also called a time-harmonic electromagnetic field. From Fourier analyses, we know that any periodic or aperiodic time function can be expressed in terms of the sum of sinusoidal functions if it satisfies certain conditions. Hence, there is justification for us to concentrate on sinusoidal electromagnetic fields.

A complex vector can be used to represent these sinusoidal quantities with the same frequency. Namely, only the amplitude and the space phase are considered, while the time dependent part is omitted. The electric field intensity can be expressed in terms of a complex vector as A sinusoidal electromagnetic field is produced by charge and current that have sinusoidal time dependence. The time dependence of the field is the same as that of the source in a linear medium. Thus the field and the source of a sinusoidal electromagnetic field have the same frequency. The original instantaneous vector can be expressed in terms of the complex vector as

In practice, the measured values are usually effective values of the sinusoidal quantity, and it is denoted as . Then where Hence The complex vector is a function of space only, and it is independent of time. In addition, this complex vector representation and the operational method that makes it possible are only applicable to sinusoidal functions of the same frequency.

Hence, can be expressed as or 9. Complex Maxwell’s Equations The field and the source of a sinusoidal electromagnetic field have the same frequency in a linear medium. Hence Maxwell’s equations can be expressed in terms of complex vectors. Consider the derivative of a sinusoidal function with respect to time tsuch that

and Since the above equation holds for any moment t, the imaginary symbol can be eliminated. In the same way, we find The above equations are called Maxwell’s equations in the frequency domain, with all field variables being complex.

Due to Since the electric field has only the y-component, and it is independent of the variable y, so that , then We find Example. The instantaneous value of the electric field intensity of a time-variable electromagnetic field in vacuum is Find complex magnetic field intensity. Solution: From the instantaneous value we have the complex vector of the electric field intensity as

Consider the time delay factor , for a sinusoidal function it leads to a phase delay of . Let Then 10. Complex Potentials The equations for the potential functions can be expressed in terms of the complex vector as

We obtain The complex Lorentz condition is The complex electric and magnetic fields can be expressed in terms of the complex potentials as

where and are the conjugates of the complex vectors and , respectively. 11. Complex Energy Density and Energy Flow Density Vector The instantaneous energy densities of electric and magnetic origins, respectively, are The corresponding maximum values Or in complex vector as

i.e. Which can be rewritten as Or in terms of the maximum value as or The effective value of a sinusoidal quantity is the root-mean-square of the instantaneous value. Hence, the average over a period value of the energy densities of a sinusoidal electromagnetic field is

which states that the average value of the energy density of a sinusoidal electromagnetic field is equal to half of the sum of the maximum values of the electric energy density and the magnetic energy density. Similarly, the power dissipation per unit volume can also be expressed in terms of the complex vector, and the maximum value is The time average is The time average of the power dissipation per unit volume is also the half of the maximum value.

The instantaneous value of the energy flow density vector Sis The time average is Let Where Sc a complex energy flow density vector, and and are the effective values. It can be written in terms of the maximum value as

Chapter 7 Time-varying Electromagnetic Fields