4th Maxwell Equation
The 4th Maxwell Equation, also known as Gauss's law for magnetism, states that the magnetic field lines are continuous, with no beginning or end, and that there are no magnetic monopoles (isolated magnetic charges). This equation helps to describe the behavior of magnetic fields and their relationship to electric currents and charges.
7 Key excerpts on "4th Maxwell Equation"
eBook - ePubLiquid Crystal Displays
Fundamental Physics and Technology
- Robert H. Chen(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...So Maxwell postulated an addition to Ampere’s law, and called it the electric displacement current (later to appear in his field equations as the electric displacement vector D), which was manifested by a change in the net flux of the electric field. This electric displacement current would provide the magnetic field elasticity and the total electric current needed to comport with experiment. With the addition of the displacement current, the magnetic circulation now becomes and inserting the expression for the displacement current given above, Ampere’s law then becomes what later came to known as the “Ampere–Maxwell law”, The derivative form of the above Ampere–Maxwell law is just Maxwell’s fourth equation, and as might be expected, the derivation again uses Stokes’ theorem, this time for the magnetic induction vector (B), Substituting the right hand term of Stokes’ theorem into the Ampere–Maxwell law equation gives, Because electromagnetic fluxes are generally continuous and differentiable in the mathematical curve sense, the derivative operator can be brought into the integrand and the surface integral brought outside, so and since the definition of electric current density is the first surface integral on the right-hand side is just J, and again, since the surfaces are arbitrarily chosen, the integrands are equal, but because the spatial change of the electric field vector E must be considered, the single-variable differential operator becomes the partial differential operator, so The equation above is just Maxwell’s fourth equation in differential form. The continuously self-circulating magnetic field is described by its curl, and the change in the electric field produces a current denoted by its current density J ; and conversely, the electric current and the change in the electric field produce the concentric curling magnetic field...
eBook - ePub- Wen Geyi(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...It must be mentioned that the above vectorial form of Maxwell equations is due to the English engineer Oliver Heaviside (1850–1925), and is presented with neatness and clarity compared to the large set of scalar equations proposed by Maxwell. Maxwell equations are the starting point for the investigation of all macroscopic electromagnetic phenomena. In (1.21), r is the observation point of the fields in meters and t is the time in seconds; H is the magnetic field intensity measured in ampères per meter (A/m); B is the magnetic induction intensity measured in tesla (N/A-m); E is electric field intensity measured in volts per meter (V/m); D is the electric induction intensity measured in coulombs per square meter (C/m 2); J is electric current density measured in ampères per square meter (A/m 2); ρ is the electric charge density measured in coulombs per cubic meter (C/m 3). The first equation is Ampère’s law, and it describes how the electric field changes according to the current density and magnetic field. The second equation is Faraday’s law, and it characterizes how the magnetic field varies according to the electric field. The minus sign is required by Lenz’s law, that is, when an electromotive force is generated by a change of magnetic flux, the polarity of the induced electromotive force is such that it produces a current whose magnetic field opposes the change, which produces it. The third equation is Coulomb’s law, and it says that the electric field depends on the charge distribution and obeys the inverse square law. The final equation shows that there are no free magnetic monopoles and that the magnetic field also obeys the inverse square law. It should be understood that none of the experiments had anything to do with waves at the time when Maxwell derived his equations. Maxwell equations imply more than the experimental facts...
eBook - ePub- Dennis H. Goldstein(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...27 Maxwell’s Equations for Electromagnetic Fields Maxwell’s equations describe the basic laws of the electromagnetic field. Over the 40 years preceding Maxwell’s enunciation of his equations (1865), the four fundamental laws describing the electromagnetic field had been discovered. They are known as Ampère’s Law, Faraday’s Law, Coulomb’s Law, and the magnetic continuity law. These four laws were cast by Maxwell, and further refined by his successors, into four differential equations: ∇ × H = j + ∂ D ∂ t, (27.1) ∇ × E = − ∂ B ∂ t, (27.2) ∇ ⋅ D = ρ, (27.3) ∇ ⋅ B = 0. (27.4) These are Maxwell’s famous equations for fields and sources in macroscopic media: E and H are the instantaneous electric and magnetic fields, D and B are the displacement vector and the magnetic induction vector, and j and ρ are the current and the charge density, respectively. We note that Equation 27.1 without the term ∂D /∂t is Ampère’s Law; the second term in Equation 27.1 was added by Maxwell and is called the displacement current. A very thorough and elegant discussion of Maxwell’s equations is given in the text Classical Electrodynamics by J. D. Jackson, and the reader will find the required background to Maxwell’s equations there [ 1 ]. When Maxwell first arrived at his equations, the term ∂D/∂t was not present. He added this term because he observed that Equation 27.1 did not satisfy the continuity equation. To see that the addition of this term leads to the continuity equation, we take the divergence of both sides of Equation 27.1, thus ∇ ⋅ [ ∇ × H ] = (∇ ⋅ j) + ∂ ∂ t (∇ ⋅ D). (27.5) The divergence of the curl is zero, the left-hand side is zero, and we have (∇ ⋅ j) + ∂ ∂ t + (∇ ⋅ D) = 0. (27.6) Next, we substitute Equation 27.3 into Equation 27.6 and find...
- G. Jagadeeswar Reddy, T. Jayachandra Prasad(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...This is called Lenz’s law. 6. Write Maxwell’s equation for time varying fields. ∇ × E ¯ = − ∂ B ¯ ∂ t 7. Name the sources of generating emf produced fields. Electric generator, Batteries and Fuel cells etc. 8. What is the In-Consistency of Ampere’s Law? We know the differential form of Ampere’s law i.e., ∇ × H ¯ = J ¯. where J ¯ is the conduction current density By taking divergence on both sides of the above equation ∇ ⋅ (∇ × H ¯) = ∇ ⋅ J ¯ We know the identity ∇ ⋅ (∇ × A ¯) = 0, where Ā is any. vector ∴ ∇ ⋅ J ¯ = 0 (a) According to Continuity equation for time varying fields ∇ ⋅ J ¯ = − ∂ ρ V ∂ t (b) For time varying fields equations (a) and (b) are not comparable. Hence Ampere’s law can not be used directly for time varying fields. To overcome inconsistency of Ampere’s law, modify the differential form of Ampere’s law as ∇ × H ¯ = J ¯ + J ¯ D where J ¯ D is the displacement current density. 9. What is the modified Maxwell’s equation for time varying fields? ∇ × H ¯ = J ¯ + ∂ D ¯ ∂ t 10. What is the ratio of Displacement to Current Conduction Current? D i s p l a c e m e n t current C o n d u c t i o n current = I D I C = ω ∊ σ 11. Write word form of Ampere’s circuit. Law. The magneto motive force around a closed path is equal to the sum of conduction current density and the time derivative of electric flux density through a surface i.e., bounded by path. 12. Write word form of Faraday’s Law. The electromotive force around a closed path is equal to the time derivative of magnetic flux density through a surface i.e., bounded by path. 13. Write word form of Gauss’s Law. The electric flux through a closed surface is equal to the charge enclosed by that surface. 14. Write word form of Gauss’s Law for magnetic fields. The magnetic flux through a closed surface is zero...
eBook - ePub- Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...In so doing it will be assumed that the reader has studied (a) the operations and functions of vector analysis (key definitions and formulas are summarized in Appendix A) and (b) the basic concepts of static and quasistatic electric and magnetic field theory. The latter will be recapitulated here in order to provide a logical base for applications of Maxwell's equations and the accompanying boundary conditions. The following elementary phenomena indicate the space pervasiveness of electromagnetic effects: The presence of current in two circuits in proximity to each other is accompanied by mechanical forces on each conductor, forces which change if either current is changed. A changing of the current in either of two such circuits is accompanied by an induced voltage in the other. Capacitors consisting of metallic spheres or other conducting bodies suspended in vacuum or in an insulating medium may be charged and later discharged. During these processes wire-borne current flows onto one sphere and off the other. The presence of electric charges on two bodies is accompanied by a mechanical force on each, forces which change if either charge is changed. Mechanical-force effects, items 1 and 4, have been mentioned primarily because they help in assigning directions to the fields. Item 2 states an application of Faraday's law, and item 3 describes a situation in which the displacement current, postulated by Maxwell, complements a discontinuous conduction current to yield, as a resultant, a composite current which is continuous. a. Directional Properties of the Electric and Magnetic Fields The direction to be assigned to a vector field is an important part of the definition of the function and is chosen in the light of the physical situation to which it relates...
eBook - ePubFields of Force
The Development of a World View from Faraday to Einstein.
- William Berkson(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...As mentioned above, the force between two current elements in Ampère’s theory is proportional to the strengths of the two currents and to the inverse square of the distance between the current elements—in analogy to Newton’s law of gravitation. The analogy to Newton breaks down because the force (a central force) is also a function of the angles between the two elements. In order to calculate the force between two currents (A and B) on Ampère’s theory, we must first add or integrate the effect of each current element of A on a particular element of B. If we thus work out the effect of A on each elment of B, and add the effects, we get the force of one current upon another. Ampère insisted on not applying his theory to open currents, as he was unable to carry out any experiments with them. Neumann thought that the induced currents must be some function of the already known ‘Ampèrian’ forces, and set out to find that function. 2 He was inspired in his idea by the theory of E. Lenz. Lenz had noticed that the forces due to induced currents were always such as to oppose the change in the forces existing before induction. 3 For example, if we have two like currents (which attract) and move them closer to each other, then the current induced will tend to decrease the attraction between the wires. By Ampère’s theory, the act of moving them closer together tends to increase the attraction. What Neumann discovered was that this mathematical function, giving the ‘electromotive’ forces causing the induced current, was the rate of change of the ‘potential’ of the force of one current upon the other. The ‘potential’ function was already a well-known function and the forces themselves were already calculable from Ampère’s theory...
eBook - ePubElectromagnetics Explained
A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
- Ron Schmitt(Author)
- 2002(Publication Date)
- Newnes(Publisher)
...(The DC magnetic field can be supplied by permanent magnets or by an electromagnet.) With a DC field applied, ferrites become anisotropic; that is, their magnetic properties are different in different directions. Simply stated, the DC field causes the ferrite to be saturated in the direction of the field while remaining unsaturated in the other two directions. Voltage-controlled phase-shifters and filters as well as exotic directional devices such as gyrators, isolators, and circulators can be created with ferrites in the microwave region. MAXWELL’S EQUATIONS AND THE DISPLACEMENT CURRENT In the 1860s, the British physicist James Clerk Maxwell set himself to the task of completely and concisely writing all the known laws of electricity and magnetism. During this exercise, Maxwell noticed a mathematical inconsistency in Ampere’s law. Recall that Ampere’s law predicts that a magnetic field surrounds all electric currents. To fix the problem, Maxwell proposed that not only do electric currents, the movement of charge, produce magnetic fields, but changing electric fields also produce magnetic fields. In other words, you do not necessarily need a charge to produce a magnetic field. For instance, when a capacitor is charging, there exists a changing electric field between the two plates. When an AC voltage is applied to a capacitor, the constant charging and discharging leads to current going to and from the plates. As I discussed in Chapter 1, although no current ever travels between the plates, the storing of opposite charges on the plates gives the perceived effect of a current traveling through the capacitor. This virtual current is called displacement current, named so because the virtual current arises from the displacement of charge at the plates...
