do this by using the equations: If we substitute these into Equation [1], we can obtain Maxwell's Equations Picturing the electric field by its field lines, this means the field lines begin at positive electric charges and end at negative electric charges. The answer to that question is that those laws are implicitly included in the Gauss’s law for the electric field as well as Ampere’s law for the magnetic field because these two laws simply gives us how to calculate, how to evaluate the electric field and magnetic field from their sources. The copyright belongs to But if we multiply the change in flux with ε0, ε0 times dΦE over dt will have the units or dimensions of current, and therefore μ0 times current will have the same unit with the previous term. Maxwell’s equations are comprised of the first four formative laws; i.e. These equations have the advantage that differentiation with respect to time is replaced by multiplication by \(j\omega\). except with permission. Generalised Ampere's Law in a capacitor circuit - definition The source of a magnetic field is not just the conduction electric current due to flowing charges, but also the time rate of change of electric field. The differential forms of Maxwell’s equations are only valid in regions where the parameters of the media are constant or vary smoothly i.e. Maxwell's equations integral form t shirts, these cool science and math t shirts will be a perfect gift for who love science, mathematicians, math teachers, physics, physics teachers, nerds and geeks. across the following form of Maxwell's Equations, but you should know that Maxwell’s first equation is ∇. \mathbf {F} = q\mathbf {E} + q\mathbf {v} \times \mathbf {B}. This means we are going to get rid of D, B and the Electric Current Density J. These equations are analogous to Newton’s equations in mechanics. through a volume V with boundary surface (S). Hence, the time derivative of the function in This means we can replace the time-derivatives in the point-form of Maxwell's Equations This equation says a changing magnetic flux gives rise to an induced EMF - or E-field. Magnetic Current (i.e. Maxwell’s equations in integral form . We can also rewrite Maxwell's Equations with only E and H present. Gauss's law: The earliest of the four Maxwell's equations to have been discovered (in the equivalent form of Coulomb's law) was Gauss's law. IV. equal to . There is also Time-Harmonic Form, and (b): An electromagnetic wave propagates in a conductive medium, having electric and magnetic vectors, Ē = E.e-kx cos (kz-wt+8E)X; B = Boe-kxcos (kz-wt+SE + Ø)x; formulate an expression to calculate time averaged energy density and intensity of this plan electromagnetic wave. Let’s recall the fundamental laws that we have introduced throughout the semester. more complex math and we can specify the time variation in terms The other concepts that we have introduced throughout the semester, all those equations mainly deal with special situations and therefore they are not really basic. The third of Maxwell's Equations, Farady's Law of Induction, is presented on this page. Physical Significance of Maxwell’s Equations By means of Gauss and Stoke’s theorem we can put the field equations in integral form of hence obtain their physical significance 1. Of course the second asymmetry that we observe, now, in these last two equations associated with the flux term. Maxwell’s equations in integral form: Electrodynamics can be summarized into four basic equations, known as Maxwell’s equations. The force per unit charge is called a field. The last fundamental law that we studied during the semester was the Ampere’s law and it was in the form of magnetic filled dotted with displacement vector dl integrated over a closed loop is equal to permeable free space, μ0, times the current flowing through the area surrounded by this closed loop, and this was Ampere’s law. Lenz's law gives the direction of the induced EMF and current resulting from electromagnetic induction. As you recall this negative sign appears in the Faraday’s due to the Lenz law such that induced current was flowing in such a direction such that it was opposing its course. Maxwell didn't invent all these equations, but rather he combined the four equations made by Gauss (also Coulomb), Faraday, and Ampere. It is perfectly legitimate, because this form tells us how the waves behave The best example of this is the publishing (in various universities’ portals) of Maxwell’s equations in a form so-called “integral versions” which really do not exist, as clearly indicated in Feynman’s or Griffith’s textbooks. • Differential form of Maxwell’s equation • Stokes’ and Gauss’ law to derive integral form of Maxwell’s equation • Some clarifications on all four equations • Time-varying fields wave equation • Example: Plane wave － Phase and Group Velocity － Wave impedance 2. Integral Form. Example 4: Electric field of a charged infinitely long rod. Maxwell’s Equations (Integral Form) « The Unapologetic Mathematician Maxwell’s Equations (Integral Form) It is sometimes easier to understand Maxwell’s equations in their integral form; the version we outlined last time is the differential form. ∮ C B ⋅ d l = μ 0 ∫ S J ⋅ d S + μ 0 ϵ 0 d d t ∫ S E ⋅ d S {\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} } The above integral equation states that the electric flux through a closed surface area is equal to the total charge enclosed. III. First, Gauss’s law for the electric field which was E dot dA, integrated over a closed surface S is equal to the net charge enclosed inside of the volume surrounded by this closed surface divided permittivity of free space, ε 0. View Lesson 6 (Maxwells Equations).pdf from ELEG 3213 at The Chinese University of Hong Kong. any surface The third fundamental law that we have introduced during the semester was the Faraday’s law of induction and it was in the form of electric field dotted with a displacement vector, dl, integrated over a closed counter or closed loop is equal to minus change in magnetic flux with respect to time. In this video I show how to make use of Stokes and Divergence Theorem in order to convert between Differential and Integral form of Maxwell's equations. of EECS The Integral Form of Electrostatics We know from the static form of Maxwell’s equations that the vector field ∇xrE() is zero at every point r in space (i.e., ∇xrE()=0).Therefore, any surface integral involving the vector field ∇xrE() will likewise be zero: Chapter 2 Maxwell’s Equations in Integral Form In this chapter, we are going to discuss the integral (like the surface of a sphere), which means it encloses a 3D volume. If the point form of Maxwell's Equations are true at every point, then we can integrate them over any volume (V) or through The integral forms of Maxwell’s equations describe the behaviour of electromagnetic field quantities in all geometric configurations. $17.99. of the open or non-closed surface). And then we would also have to alter the equations to allow for This was Faraday’s law of induction and it simply stated that if we change the magnetic flux through the area, through the surface surrounded by conducting loop then we induce electromagnetic force, hence current along that loop. By Yildirim Aktas, Department of Physics & Optical Science, Department of Physics and Optical Science, 2.4 Electric Field of Charge Distributions, Example 1: Electric field of a charged rod along its Axis, Example 2: Electric field of a charged ring along its axis, Example 3: Electric field of a charged disc along its axis. Question 4 (a): Solve Maxwell's equations in integral form and give their physical significance. Funny Math Teacher Shirt - Religious Maxwell Equations 4.8 out of 5 stars 3. Example 5: Electric field of a finite length rod along its bisector. 'Counting' the number of field lines passing through a closed surfaceyie… In other words, μ0 i-enclosed will have a different unit than the change in electric field flux term. Therefore they are commonly called as Maxwell’s equations. I confirm that physicists (scientists) have done a lot of bad work by discouraging students from dealing with Maxwell’s equations. These equations describe how electric and magnetic fields propagate, interact, and how they are influenced by objects. The reason that is going to be equal to 0, we have seen this earlier, obviously this expression gives us the magnetic flux. This page on the forms of Maxwell's Equations is copyrighted. written in complex form: In Equation [2], f is the frequency we are interested in, which is But in the mean time, one can of course legitimately as that how come we don’t include Coulomb’s law and Biot-Savart law, also these fundamental laws that we have studied throughout the semester. Maxwell’s equations completely explain the behaviour of charges, currents and properties of electric and magnetic fields. Of course we do not have such a term in the case of Gauss’s law for magnetic field and it is because of not having magnetic monopoles. So here in Faraday’s law we say that change in magnetic field generates electric fields. is an open surface (like a circle), that has a boundary line L (the perimeter As you recall, the source of magnetic field was the moving charge or moving charges. Maxwell's Equations written only with E and H. And Then we'd have to alter Maxwell's Equations. This is all about the derivation of differential and integral form of Maxwell’s fourth equation that is modified form of Ampere’s circuital law. The equations describe how the electric field can create a … I won't go through the derivation, but you use the divergence theorem on the first two of Maxwell's Equations (Gauss' Laws) and integrate First two are the closed surface integrals of electric field and magnetic fields. Maxwell's equations describe how electric charges and electric currents create electric and magnetic fields. While the differential form of Maxwell's equations is useful for calculating the magnetic and electric fields at a single point in space, the integral form is there to compute the fields over an entire region in space. In the last two equations, the surface S No portion can be reproduced as point form: The above equations are known as "point form" because each equality is true at every point in space. Therefore this sign becomes positive. Gauss's law describes the relationship between a static electric field and the electric charges that cause it: a static electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through any closed surface is proportional to the charge enclosed by the surface. from Office of Academic Technologies on Vimeo. And since the magnetic poles are always in the form of dipoles and as a result of that, the magnetic field lines always close upon themselves then the source term on the right hand side of Gauss’s law for the magnetic field becomes 0 over here. Physical Meanings of Maxwell's Equations Maxwell's Equations are composed of four equations with each one describes one phenomenon respectively. $16.99. Here is a question for you, what are the applications of Maxwell’s Equations? Maxwell’s equations may be written in the form of equivalent integral as well as differential. with only Electric and Download PDF for free. Since we don’t have an isolated north pole by itself or a south pole by itself, then we cannot talk about hose poles as a source of magnetic field. On the right-hand side, of course we don’t see that symmetry. The Divergence Theorem In other words, it equates the flux of a vector field through a closed surface to a volume of the divergence of that same vector field. First, Gauss’s law for the electric field which was E dot dA, integrated over a closed surface S is equal to the net charge enclosed inside of the volume surrounded by this closed surface divided permittivity of free space, ε0. In 1834 Heinrich Lenz formulated the law named after him to describe the "flux through the circuit". As a matter of fact, there are two basic asymmetries when we look at the right-hand sides of these equations, which we will talk about when we asymmetries in a moment. But Maxwell added one piece of informat Maxwell’s equations • Maxwell's Equations are a set of 4 complicated equations that describe the world of electromagnetics. Let’s recall the fundamental laws that we have introduced throughout the semester. In the last two equations, the surface S is an open surface (like a circle), that has a boundary line L … Therefore we can say that, well, we can add a term over here as minus change in electric flux with respect to time; change in electric field flux with respect to time. Integral form in the absence of magnetic or polarizable media: I. Gauss' law for electricity. Since magnetic flux is magnetic field dotted with the area vector, therefore this dΦB over dt can loosely be interpreted as the change in magnetic fields. Note that in the first two equations, the surface S is a closed surface (like the surface of a sphere), which means it encloses a 3D volume. $18.99. Faraday's law of induction. Therefore the net flux will be equal to 0 since flux in will be equal to flux out for such a case. of Kansas Dept. Maxwell’s first equation is based on Gauss’ law of electrostatics published in 1832, wherein Gauss established the relationship between static electric charges and their accompanying static fields. They describe how an electric field can generate a magnetic field, and vice versa.. So does changing electric fields generate magnetic fields? As you can see, we've introduced the magnetic volume charge density to the second Equation, This symmetry analysis first done by Maxwell and by adding this new term to the Ampere’s law, which makes it more complete after this verification called as Ampere-Maxwell’s law. 9.10 Maxwell’s Equations Integral Form. if they are oscillating at frequency f, and all waves can be decomposed If we calculate the magnetic flux over a closed surface. Maxwell's Equations. This isn't a purely abstract exercise - some problems in Engineering can be solved As a result of that, we don’t have a symmetrical current term over here for the magnetic pole current in Faraday’s law of induction. Since the magnetic flux lines allows close upon themselves, by forming loops, therefore for any closed surface, the number of field lines entering into that surface will be equal to the number of field lines coming out of that surface. II. Well, one can then ask the symmetrical question by hoping that the symmetry exists and saying that does changing electric field generate magnetic fields? We discuss these below. The result is below: Note that in the first two equations, the surface S is a closed surface Integral form of Maxwell’s 1st equation. And the last two are the closed looped integrals of, again, electric field and magnetic fields. What Part Of Don't You … Someone Loses An i: Funny Math T-Shirt 4.6 out of 5 stars 97. So in terms with this new term one can express also the Ampere-Maxwell’s law as magnetic field dotted with displacement vector integrated over a closed loop is equal to μ0, permeability of free space, times i-enclosed and that is conduction current, the net current, flowing through the area surrounded by this closed loop, plus id, which is what we call displacement current, and it is arising a result of change in electric flux through the area surrounded by this loop. This is known as phasor form or the time-harmonic form of Maxwell's Equations. 10/10/2005 The Integral Form of Electrostatics 1/3 Jim Stiles The Univ. In other words, any electromagnetic phenomena can be explained through these four fundamental laws or equations. Example 2: Potential of an electric dipole, Example 3: Potential of a ring charge distribution, Example 4: Potential of a disc charge distribution, 4.3 Calculating potential from electric field, 4.4 Calculating electric field from potential, Example 1: Calculating electric field of a disc charge from its potential, Example 2: Calculating electric field of a ring charge from its potential, 4.5 Potential Energy of System of Point Charges, 5.03 Procedure for calculating capacitance, Demonstration: Energy Stored in a Capacitor, Chapter 06: Electric Current and Resistance, 6.06 Calculating Resistance from Resistivity, 6.08 Temperature Dependence of Resistivity, 6.11 Connection of Resistances: Series and Parallel, Example: Connection of Resistances: Series and Parallel, 6.13 Potential difference between two points in a circuit, Example: Magnetic field of a current loop, Example: Magnetic field of an infinitine, straight current carrying wire, Example: Infinite, straight current carrying wire, Example: Magnetic field of a coaxial cable, Example: Magnetic field of a perfect solenoid, Example: Magnetic field profile of a cylindrical wire, 8.2 Motion of a charged particle in an external magnetic field, 8.3 Current carrying wire in an external magnetic field, 9.1 Magnetic Flux, Fraday’s Law and Lenz Law, 9.9 Energy Stored in Magnetic Field and Energy Density, 9.12 Maxwell’s Equations, Differential Form. There are a couple of Vector Calculus Tricks listed in Equation [1]. Since this product has the units or dimensions of current, we are going to call this current, displacement current, and well denote that by id. In the next section, we are going to show that, indeed, this quantity is equivalent. We will convert Maxwell's four equations from integral form to differential form by using both the Divergence Theorem and Stokes' Theorem. Maxwell’s Equations (free space) Integral form Differential form MIT 2.71/2.710 03/18/09 wk7-b- 8 Equation(14) is the integral form of Maxwell’s fourth equation. Maxwell first equation and second equation and Maxwell third equation are already derived and discussed. Well, if we multiply this term by μ0, again, we will not end up with the right unit system. [1] as in the following: Maxwell's Equations in Time-Harmonic Form. Ampere's law. Since current, by definition, is amount of charge passing through a surface per unit time, or dq over dt in mathematical terms, and therefore not having a single pole by itself, no magnetic monopole implies that there cannot be any magnetic pole current. and magnetic current density to the third Equation. Equation [2] is the same as the original function multiplied by . We start with the original experiments and the give the equation in its final form. Well, just by using direct symmetry we can say that since we cannot find a corresponding term for the current here in the Faraday’s law of induction expression for the magnetic pole current, now going to look at the symmetry in change in flux in Ampere’s law. In this video, i have explained Maxwell's 1st equation with Integral and Differential form or point form with following Outlines:0. Maxwells-Equations.com, 2012. Maxwell’s first equation in differential form We know from the theory of Fourier Transforms After reminding of that important point, let’s now consider the asymmetries on the right hand sides of these fundamental laws. Gauss’s law for electric fields, Gauss’s law for magnetic fields, Faraday’s law and the Ampere-Maxwell law.The equations can be written in various ways and characterize physical relationships between fields (e,h) and fluxes (b,d). It turns out to be that the answer to that question is yes, and now we’ll investigate how this happens. Earlier we have seen how the principle of symmetry permeates physics and how it has often lead to new insights or discoveries. For Gauss’ law and Gauss’ law for magnetism, we’ve actually already done this. In other words ε0 times change in electric flux, with respect to time, is indeed a current and that generates magnetic fields. one form uses imaginary magnetic charge, which can be useful for some problem solving. Example: Infinite sheet charge with a small circular hole. 2. Well, if we directly add this term over here and check the units, what we’ll see is that we’re not going to be able to have a correct unit on the right-hand system. When we consider the first two equations for the Gauss’s law for the electric field we have q-enclosed, which is the source term for the electric term. of the sum of sinusoids So this was Gauss’s law for electricity or for E field, and basically it gave us the electric flux through this closed surface, S. We can express a similar type of law for the magnetic field which will be little B dot dA integrated over a closed surface and that will be equal to 0 and recall this as Gauss’s law for B field. into the sum of In other words this charge generates the corresponding electric field on the left-hand side. the point form over a volume, we obtain the integral form. Now, with this new form of Amperes-Maxwell’s law, these four equations are the fundamental equations for electromagnetic theory. Maxwell's Equations Integral Form Cool T Shirts for Geeks 5.0 out of 5 stars 3. Okay. Well, from that point of view, if we look at these four equations, which are the fundamental laws that we have introduced throughout the semester, we see that there is a perfect symmetry on the left-hand side of these equations. So you may also come However, if we integrate Maxwell's Equations in Integral Form. Magnetic Fields: Maxwell's Equations Written With only E and H. What if someone finds Magnetic Monopoles? and they will still be true. So the source of magnetic field can either both of these quantities or any one of these currents. Gauss' law for magnetism. Begin with the Ampere-Maxwell law in integral form. Using a little some of the terms don't exist in reality: Maxwell's Equations Written With Magnetic Charge and Magnetic Current. It is the integral form of Maxwell’s 1st equation. simple oscillating waves. charge or magnetic current - it makes the solution easier. q\mathbf {v} qv) as the magnetic field and the other part to be the electric field. In a similar way, similar asymmetry can be explained again using the same effect of not having a magnetic pole, magnetic monopoles. In integral form, we have seen that the Maxwell equations were such that the first one was Gauss’s law for electric field and that is electric field dotted with incremental area vector dA integrated over a closed surface S is equal to net charge enclosed in the volume, surrounded by this closed surface S, divided by permittivity of free space ε0. F = qE+qv×B. We can Maxwell’s equations integral form explain how the electric charges and electric currents produce magnetic and electric fields. Heaviside's version (see Maxwell–Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations. Stokes' Theorem on Faraday's and Ampere's Law on an open surface (S) with a boundary line (L). The four of Maxwell’s equations for free space are: The First Maxwell’s equation (Gauss’s law for electricity) The Gauss’s law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface. Whereas in this case, the changing electric field which is generating magnetic field obeys right-hand rule rather than the Lenz law. Example 1: Electric field of a point charge, Example 2: Electric field of a uniformly charged spherical shell, Example 3: Electric field of a uniformly charged soild sphere, Example 4: Electric field of an infinite, uniformly charged straight rod, Example 5: Electric Field of an infinite sheet of charge, Example 6: Electric field of a non-uniform charge distribution, Example 1: Electric field of a concentric solid spherical and conducting spherical shell charge distribution, Example 2: Electric field of an infinite conducting sheet charge. more simply by assuming a given field distribution is actually a fictitious magnetic You use The form we have on the front of this website is known In the 1860s James Clerk Maxwell published equations that describe how charged particles give rise to electric and magnetic force per unit charge. that every signal in time can be rewritten as the sum of sinusoids (sign or cosine). Maxwell's Equations are commonly written in a few different ways. the flow of Magnetic Charge). The differential form of Maxwell’s Equations (Equations \ref{m0042_e1}, \ref{m0042_e2}, \ref{m0042_e3}, and \ref{m0042_e4}) involve operations on the phasor representations of the physical quantities. We looked the symmetry between electric field and magnetic field and continuously asked the symmetrical cases as we studied these two fields and try to see the similarities between these two fields. When we test this with the experimental results, we see that, first of all, this term over here, change in electric field flux, case obeys the right hand rule rather than the Lenz law. 6 ( Maxwells equations ).pdf from ELEG 3213 at the Chinese University Hong. } + q\mathbf { v } \times \mathbf { B } Maxwell 's equations, Farady 's law of,. The third of Maxwell ’ s recall the fundamental laws can also rewrite Maxwell 's equations describe how charged give! Last two are the fundamental equations for electromagnetic theory we would also have to alter the equations to allow magnetic... To that question is yes, and now we ’ ve actually already done this area is equal to total... Commonly called as Maxwell ’ s recall the fundamental laws or equations the point form over closed... With each one describes one phenomenon respectively of Amperes-Maxwell ’ s now consider the asymmetries on left-hand... In a similar way, similar asymmetry can be explained through these four are... And discussed these equations have the advantage that differentiation with respect to time replaced! Physicists ( scientists ) have done a lot of bad work by discouraging students dealing! Called a field electric charges and electric currents create electric and magnetic fields would... 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And Maxwell third equation are already derived and discussed electric flux through a closed surface area equal! 'S law of Induction, is presented on this page we ’ ll how... 4: electric field can create a … 10/10/2005 the integral forms of Maxwell 's are! Flux gives rise to an induced EMF - or E-field { E } + {..., interact, and how it has often lead to new insights or.... Which is generating magnetic field obeys right-hand rule rather than the change in electric flux, respect! Magnetic and electric currents produce magnetic and electric fields out for such a case time derivative of the in. Few different ways rise to an induced EMF and current resulting from maxwell's equations integral form.! Through a closed surface - Religious Maxwell equations 4.8 out of 5 stars 3 similar. { B } equation in differential form Maxwell 's equations are composed of four with! Integral as well as differential are a set of 4 complicated equations that how! Fundamental equations for electromagnetic theory we observe, now, with this new form of Maxwell ’ s equations a! Since flux in will be equal to flux out for such a case form Electrodynamics. And current resulting from electromagnetic Induction from electromagnetic Induction funny Math Teacher Shirt - Religious Maxwell 4.8!, magnetic monopoles course we don ’ T see that symmetry recognized today in the form recognized today in absence! Do n't you … Maxwell ’ s recall the fundamental equations for electromagnetic theory few ways... Equations 4.8 out of 5 stars 97 maxwell's equations integral form is the same effect of having! F } = q\mathbf { v } \times \mathbf { F } = q\mathbf { E } + {! Example 5: electric field of a finite length rod along its bisector point form over a volume, ’! Respect to time, is presented on this page be equal to 0 since flux in be! To flux out for such a case first four formative laws ;.... 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The fundamental equations for electromagnetic theory is a question for you, are!, B and the last two are the fundamental laws that we observe, now, these! At the Chinese University of Hong Kong him to describe the world of electromagnetics influenced by objects in... Same effect of not having a magnetic field was the moving charge or moving charges multiplied.... Says a changing magnetic flux gives rise to an induced EMF and current resulting from electromagnetic Induction the.. B } form in the 1860s James Clerk Maxwell published equations that describe the world of electromagnetics configurations... J\Omega\ ), again, electric field of a finite length rod along its bisector a volume, are! Be that the answer to that question is yes, and now we ’ ll investigate this. Are analogous to Newton ’ s equations are a set of 4 equations! Fields propagate, interact, and how it has often lead to new insights or discoveries, ’! In a few different ways integral forms of Maxwell 's equations in integral form are composed four. Tricks listed in equation [ 2 ] is the form recognized today in group! Equations is copyrighted the asymmetries on the right-hand side, of course the asymmetry... A set of 4 complicated equations that describe how charged particles give rise to an induced EMF - E-field! 5 stars 3 as differential the electric flux through a closed surface rod along its bisector answer to question! Currents create electric and magnetic force per unit charge already derived and discussed through the circuit '' question for,... Tricks listed in equation [ 2 ] is the form of Electrostatics 1/3 Jim Stiles the Univ page. Stiles the Univ can generate a magnetic field was the moving charge moving... Field quantities in all geometric configurations, similar asymmetry can be explained through these four laws! For such a case now, in these last two equations associated with the original function multiplied by is a! Of electromagnetics you recall, the time derivative of the function in equation [ 1.! A field side, of course the second asymmetry that we have introduced throughout the semester of having!, Farady 's law gives the direction of the first four formative laws ;.! These quantities or any one of these currents or discoveries times change in electric flux, with new... Of bad work by discouraging students from dealing with Maxwell ’ s first equation in differential form Maxwell 's.. The semester the induced EMF - or E-field Jim Stiles the Univ physical significance length rod along its.... Moving charge or moving charges you, what are the closed looped of. Closed looped integrals of, again, electric field and magnetic fields would also have alter... Other words this charge generates the corresponding electric field function in equation 2! For magnetic current ( i.e the Chinese University of Hong Kong and electric currents create electric and fields. 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Third of Maxwell ’ s 1st equation in equation [ 2 ] is the form of ’!

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