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82-418: Both of Kirchhoff's laws can be understood as corollaries of Maxwell's equations in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits. This law, also called Kirchhoff's first law , or Kirchhoff's junction rule , states that, for any node (junction) in an electrical circuit ,

164-459: A homotopy F : [ 0 , 1 ] × [ 0 , 1 ] → X {\displaystyle F:[0,1]\times [0,1]\to X} such that F ( x , 0 ) = p ( x ) {\displaystyle F(x,0)=p(x)} and F ( x , 1 ) = q ( x ) . {\displaystyle F(x,1)=q(x).} A topological space X {\displaystyle X}

246-541: A vector field , and the magnetic field , B , a pseudovector field, each generally having a time and location dependence. The sources are The universal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are: In the differential equations, In the integral equations, The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over

328-490: A changing magnetic field and generates an electric field in a nearby wire. The original law of Ampère states that magnetic fields relate to electric current . Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current . The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve. Maxwell's modification of Ampère's circuital law

410-406: A changing magnetic field through Maxwell's modification of Ampère's circuital law . This perpetual cycle allows these waves, now known as electromagnetic radiation , to move through space at velocity c . The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This

492-1288: A constant speed in vacuum, c ( 299 792 458  m/s ). Known as electromagnetic radiation , these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays . In partial differential equation form and a coherent system of units , Maxwell's microscopic equations can be written as ∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 ( J + ε 0 ∂ E ∂ t ) {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\end{aligned}}} With E {\displaystyle \mathbf {E} }

574-409: A continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is C ∖ { 0 } , {\displaystyle \mathbb {C} \setminus \{0\},} which is not simply connected. The notion of simple connectedness is important in complex analysis because of the following facts: The notion of simple connectedness

656-875: A factor (see Heaviside–Lorentz units , used mainly in particle physics ). The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem . According to the (purely mathematical) Gauss divergence theorem , the electric flux through the boundary surface ∂Ω can be rewritten as The integral version of Gauss's equation can thus be rewritten as ∭ Ω ( ∇ ⋅ E − ρ ε 0 ) d V = 0 {\displaystyle \iiint _{\Omega }\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\,\mathrm {d} V=0} Since Ω

738-869: A fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume. The definitions of the auxiliary fields are: D ( r , t ) = ε 0 E ( r , t ) + P ( r , t ) , H ( r , t ) = 1 μ 0 B ( r , t ) − M ( r , t ) , {\displaystyle {\begin{aligned}\mathbf {D} (\mathbf {r} ,t)&=\varepsilon _{0}\mathbf {E} (\mathbf {r} ,t)+\mathbf {P} (\mathbf {r} ,t),\\\mathbf {H} (\mathbf {r} ,t)&={\frac {1}{\mu _{0}}}\mathbf {B} (\mathbf {r} ,t)-\mathbf {M} (\mathbf {r} ,t),\end{aligned}}} where P

820-502: A fuller description. Simply connected space In topology , a topological space is called simply connected (or 1-connected , or 1-simply connected ) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such

902-772: A given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law: d d t ∬ Σ B ⋅ d S = ∬ Σ ∂ B ∂ t ⋅ d S , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {S} \,,} Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using

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984-431: A handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected . The definition rules out only handle -shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on

1066-597: A hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. A topological space X {\displaystyle X} is called simply connected if it is path-connected and any loop in X {\displaystyle X} defined by f : S 1 → X {\displaystyle f:S^{1}\to X} can be contracted to

1148-476: A negative sign which means the assumed direction of i 3 was incorrect and i 3 is actually flowing in the direction opposite to the red arrow labeled i 3 . The current in R 3 flows from left to right. Maxwell%27s equations Maxwell's equations , or Maxwell–Heaviside equations , are a set of coupled partial differential equations that, together with the Lorentz force law, form

1230-468: A point: there exists a continuous map F : D 2 → X {\displaystyle F:D^{2}\to X} such that F {\displaystyle F} restricted to S 1 {\displaystyle S^{1}} is f . {\displaystyle f.} Here, S 1 {\displaystyle S^{1}} and D 2 {\displaystyle D^{2}} denotes

1312-401: A time-varying magnetic field corresponds to curl of an electric field . In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface. The electromagnetic induction is the operating principle behind many electric generators : for example, a rotating bar magnet creates

1394-544: A wire is not an ideal conductor. Unlike an ideal conductor, wires can inductively and capacitively couple to each other (and to themselves), and have a finite propagation delay. Real conductors can be modeled in terms of lumped elements by considering parasitic capacitances distributed between the conductors to model capacitive coupling, or parasitic (mutual) inductances to model inductive coupling. Wires also have some self-inductance. Assume an electric network consisting of two voltage sources and three resistors. According to

1476-414: Is a simply connected space which maps to X {\displaystyle X} via a covering map . If X {\displaystyle X} and Y {\displaystyle Y} are homotopy equivalent and X {\displaystyle X} is simply connected, then so is Y . {\displaystyle Y.} The image of a simply connected set under

1558-557: Is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied if and only if the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement. Similarly rewriting the magnetic flux in Gauss's law for magnetism in integral form gives which is satisfied for all Ω if and only if ∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0} everywhere. By

1640-414: Is constant. Whenever the electric field between parts of the circuit is non-negligible, such as when two wires are capacitively coupled , this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable. For example, in a transmission line , the charge density in the conductor may be constantly changing. On the other hand, the voltage law relies on

1722-1265: Is equivalent to { i 1 + ( − i 2 ) + ( − i 3 ) = 0 R 1 i 1 + R 2 i 2 + 0 i 3 = E 1 0 i 1 + R 2 i 2 − R 3 i 3 = E 1 + E 2 {\displaystyle {\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}} Assuming R 1 = 100 Ω , R 2 = 200 Ω , R 3 = 300 Ω , E 1 = 3 V , E 2 = 4 V {\displaystyle {\begin{aligned}R_{1}&=100\Omega ,&R_{2}&=200\Omega ,&R_{3}&=300\Omega ,\\{\mathcal {E}}_{1}&=3{\text{V}},&{\mathcal {E}}_{2}&=4{\text{V}}\end{aligned}}}

SECTION 20

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1804-443: Is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space . The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, matches

1886-428: Is one of Maxwell's equations ). This has practical application in situations involving " static electricity ". Kirchhoff's circuit laws are the result of the lumped-element model and both depend on the model being applicable to the circuit in question. When the model is not applicable, the laws do not apply. The current law is dependent on the assumption that the net charge in any wire, junction or lumped component

1968-399: Is possible to still model such circuits using parasitic components . If frequencies are too high, it may be more appropriate to simulate the fields directly using finite element modelling or other techniques . To model circuits so that both laws can still be used, it is important to understand the distinction between physical circuit elements and the ideal lumped elements. For example,

2050-433: Is simply connected if and only if X {\displaystyle X} is path-connected and the fundamental group of X {\displaystyle X} at each point is trivial, i.e. consists only of the identity element . Similarly, X {\displaystyle X} is simply connected if and only if for all points x , y ∈ X , {\displaystyle x,y\in X,}

2132-574: Is simply connected if and only if both X {\displaystyle X} and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes an example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. A relaxation of the requirement that X {\displaystyle X} be connected leads to an exploration of open subsets of

2214-443: Is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping. The microscopic version is sometimes called "Maxwell's equations in vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive

2296-432: Is that the additional fields D and H need to be determined through phenomenological constituent equations relating these fields to the electric field E and the magnetic field B , together with the bound charge and current. See below for a detailed description of the differences between the microscopic equations, dealing with total charge and current including material contributions, useful in air/vacuum; and

2378-807: Is the polarization field and M is the magnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density ρ b and bound current density J b in terms of polarization P and magnetization M are then defined as ρ b = − ∇ ⋅ P , J b = ∇ × M + ∂ P ∂ t . {\displaystyle {\begin{aligned}\rho _{\text{b}}&=-\nabla \cdot \mathbf {P} ,\\\mathbf {J} _{\text{b}}&=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}.\end{aligned}}} If we define

2460-409: Is the total number of branches with currents flowing towards or away from the node. Kirchhoff's circuit laws were originally obtained from experimental results. However, the current law can be viewed as an extension of the conservation of charge , since charge is the product of current and the time the current has been flowing. If the net charge in a region is constant, the current law will hold on

2542-816: Is the total number of voltages measured. A similar derivation can be found in The Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits . Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the Maxwell–Faraday equation reveals that ∇ × E = − ∂ B ∂ t = 0 {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} } in

Kirchhoff's circuit laws - Misplaced Pages Continue

2624-447: Is usually less than c . In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law . In turn, that electric field creates

2706-801: The Ampère–Maxwell law , the modified version of Ampère's circuital law, in integral form can be rewritten as ∬ Σ ( ∇ × B − μ 0 ( J + ε 0 ∂ E ∂ t ) ) ⋅ d S = 0. {\displaystyle \iint _{\Sigma }\left(\nabla \times \mathbf {B} -\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)\cdot \mathrm {d} \mathbf {S} =0.} Since Σ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that

2788-629: The Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls ) over a surface it bounds, i.e. ∮ ∂ Σ B ⋅ d ℓ = ∬ Σ ( ∇ × B ) ⋅ d S , {\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\iint _{\Sigma }(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} ,} Hence

2870-524: The Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside , has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x , y and z components. The relativistic formulations are more symmetric and Lorentz invariant. For

2952-469: The electric potential (and thus voltage) can be defined in other ways, such as via the Helmholtz decomposition . In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of Faraday's law of induction (which

3034-546: The exterior of each of the components, from one terminal to another. Note that this derivation uses the following definition for the voltage rise from a {\displaystyle a} to b {\displaystyle b} : V a → b = − ∫ P a → b E ⋅ d l {\displaystyle V_{a\to b}=-\int _{{\mathcal {P}}_{a\to b}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} } However,

3116-446: The magnetization M . The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of P and M , which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, Maxwell's macroscopic equations ignore many details on

3198-629: The old SI system of units, the values of μ 0 = 4 π × 10 − 7 {\displaystyle \mu _{0}=4\pi \times 10^{-7}} and c = 299 792 458   m/s {\displaystyle c=299\,792\,458~{\text{m/s}}} are defined constants, (which means that by definition ε 0 = 8.854 187 8... × 10 − 12   F/m {\displaystyle \varepsilon _{0}=8.854\,187\,8...\times 10^{-12}~{\text{F/m}}} ) that define

3280-448: The speed of light ; indeed, light is one form of electromagnetic radiation (as are X-rays , radio waves , and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics . In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature ,

3362-548: The unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X {\displaystyle X} is simply connected if and only if it is path-connected, and whenever p : [ 0 , 1 ] → X {\displaystyle p:[0,1]\to X} and q : [ 0 , 1 ] → X {\displaystyle q:[0,1]\to X} are two paths (that is, continuous maps) with

Kirchhoff's circuit laws - Misplaced Pages Continue

3444-481: The vacuum permeability . The equations have two major variants: The term "Maxwell's equations" is often also used for equivalent alternative formulations . Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem , analytical mechanics , or for use in quantum mechanics . The covariant formulation (on spacetime rather than space and time separately) makes

3526-1085: The Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary: In particular, in an isolated system the total charge is conserved. In a region with no charges ( ρ = 0 ) and no currents ( J = 0 ), such as in vacuum, Maxwell's equations reduce to: ∇ ⋅ E = 0 , ∇ × E + ∂ B ∂ t = 0 , ∇ ⋅ B = 0 , ∇ × B − μ 0 ε 0 ∂ E ∂ t = 0. {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0,&\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0,\\\nabla \cdot \mathbf {B} &=0,&\nabla \times \mathbf {B} -\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}=0.\end{aligned}}} Taking

3608-611: The Gaussian ( CGS ) units. Using these definitions, colloquially "in Gaussian units", the Maxwell equations become: The equations simplify slightly when a system of quantities is chosen in the speed of light, c , is used for nondimensionalization , so that, for example, seconds and lightseconds are interchangeable, and c = 1. Further changes are possible by absorbing factors of 4 π . This process, called rationalization, affects whether Coulomb's law or Gauss's law includes such

3690-556: The ampere and the metre. In the new SI system, only c keeps its defined value, and the electron charge gets a defined value. In materials with relative permittivity , ε r , and relative permeability , μ r , the phase velocity of light becomes v p = 1 μ 0 μ r ε 0 ε r , {\displaystyle v_{\text{p}}={\frac {1}{\sqrt {\mu _{0}\mu _{\text{r}}\varepsilon _{0}\varepsilon _{\text{r}}}}},} which

3772-422: The atoms, most notably their electrons . The connection to angular momentum suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These bound currents can be described using

3854-412: The boundaries of the region. This means that the current law relies on the fact that the net charge in the wires and components is constant. A matrix version of Kirchhoff's current law is the basis of most circuit simulation software , such as SPICE . The current law is used with Ohm's law to perform nodal analysis . The current law is applicable to any lumped network irrespective of the nature of

3936-1077: The closed circuit s 2 gives: − R 3 i 3 − E 2 − E 1 + R 2 i 2 = 0 {\displaystyle -R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0} This yields a system of linear equations in i 1 , i 2 , i 3 : { i 1 − i 2 − i 3 = 0 − R 2 i 2 + E 1 − R 1 i 1 = 0 − R 3 i 3 − E 2 − E 1 + R 2 i 2 = 0 {\displaystyle {\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}} which

4018-471: The compatibility of Maxwell's equations with special relativity manifest . Maxwell's equations in curved spacetime , commonly used in high-energy and gravitational physics , are compatible with general relativity . In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences. The publication of

4100-914: The curl (∇×) of the curl equations, and using the curl of the curl identity we obtain μ 0 ε 0 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 , μ 0 ε 0 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}} The quantity μ 0 ε 0 {\displaystyle \mu _{0}\varepsilon _{0}} has

4182-401: The defining relations above to eliminate D , and H , the "macroscopic" Maxwell's equations reproduce the "microscopic" equations. In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and the electric field E , as well as the magnetizing field H and the magnetic field B . Equivalently, we have to specify

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4264-422: The dependence of the polarization P (hence the bound charge) and the magnetization M (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called constitutive relations . For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for

4346-399: The differential version and using Gauss and Stokes formula appropriately. The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε 0 and μ 0 into the units (and thus redefining these). With a corresponding change in the values of the quantities for the Lorentz force law this yields

4428-960: The dimension (T/L) . Defining c = ( μ 0 ε 0 ) − 1 / 2 {\displaystyle c=(\mu _{0}\varepsilon _{0})^{-1/2}} , the equations above have the form of the standard wave equations 1 c 2 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 , 1 c 2 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}} Already during Maxwell's lifetime, it

4510-457: The electric field, B {\displaystyle \mathbf {B} } the magnetic field, ρ {\displaystyle \rho } the electric charge density and J {\displaystyle \mathbf {J} } the current density . ε 0 {\displaystyle \varepsilon _{0}} is the vacuum permittivity and μ 0 {\displaystyle \mu _{0}}

4592-1237: The equations depend only on the free charges Q f and free currents I f . This reflects a splitting of the total electric charge Q and current I (and their densities ρ and J ) into free and bound parts: Q = Q f + Q b = ∭ Ω ( ρ f + ρ b ) d V = ∭ Ω ρ d V , I = I f + I b = ∬ Σ ( J f + J b ) ⋅ d S = ∬ Σ J ⋅ d S . {\displaystyle {\begin{aligned}Q&=Q_{\text{f}}+Q_{\text{b}}=\iiint _{\Omega }\left(\rho _{\text{f}}+\rho _{\text{b}}\right)\,\mathrm {d} V=\iiint _{\Omega }\rho \,\mathrm {d} V,\\I&=I_{\text{f}}+I_{\text{b}}=\iint _{\Sigma }\left(\mathbf {J} _{\text{f}}+\mathbf {J} _{\text{b}}\right)\cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} .\end{aligned}}} The cost of this splitting

4674-412: The equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics . Gauss's law describes

4756-761: The exterior region. If each of the components has a finite volume, then the exterior region is simply connected , and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that ∑ i V i = − ∑ i ∫ P i E ⋅ d l = ∮ E ⋅ d l = 0 {\displaystyle \sum _{i}V_{i}=-\sum _{i}\int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0} where P i {\textstyle {\mathcal {P}}_{i}} are paths around

4838-437: The fact that the actions of time-varying magnetic fields are confined to individual components, such as inductors. In reality, the induced electric field produced by an inductor is not confined, but the leaked fields are often negligible. The lumped element approximation for a circuit is accurate at low frequencies. At higher frequencies, leaked fluxes and varying charge densities in conductors become significant. To an extent, it

4920-551: The first law: i 1 − i 2 − i 3 = 0 {\displaystyle i_{1}-i_{2}-i_{3}=0} Applying the second law to the closed circuit s 1 , and substituting for voltage using Ohm's law gives: − R 2 i 2 + E 1 − R 1 i 1 = 0 {\displaystyle -R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0} The second law, again combined with Ohm's law, applied to

5002-1625: The fluid is the curl of the velocity field. The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the Ampère–Maxwell law has zero divergence by the div–curl identity . Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives: 0 = ∇ ⋅ ( ∇ × B ) = ∇ ⋅ ( μ 0 ( J + ε 0 ∂ E ∂ t ) ) = μ 0 ( ∇ ⋅ J + ε 0 ∂ ∂ t ∇ ⋅ E ) = μ 0 ( ∇ ⋅ J + ∂ ρ ∂ t ) {\displaystyle 0=\nabla \cdot (\nabla \times \mathbf {B} )=\nabla \cdot \left(\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {E} \right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}\right)} i.e., ∂ ρ ∂ t + ∇ ⋅ J = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.} By

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5084-432: The foundation of classical electromagnetism , classical optics , electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges , currents , and changes of the fields. The equations are named after

5166-413: The integrand is zero if and only if the Ampère–Maxwell law in differential equations form is satisfied. The equivalence of Faraday's law in differential and integral form follows likewise. The line integrals and curls are analogous to quantities in classical fluid dynamics : the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of

5248-424: The macroscopic equations, dealing with free charge and current, practical to use within materials. When an electric field is applied to a dielectric material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons move a tiny distance in the opposite direction. This produces a macroscopic bound charge in

5330-416: The macroscopic properties of bulk matter from its microscopic constituents. "Maxwell's macroscopic equations", also known as Maxwell's equations in matter , are more similar to those that Maxwell introduced himself. In the macroscopic equations, the influence of bound charge Q b and bound current I b is incorporated into the displacement field D and the magnetizing field H , while

5412-463: The magnetic field of a material is attributed to a dipole , and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field . The Maxwell–Faraday version of Faraday's law of induction describes how

5494-425: The material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive bound charge on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the polarization P of

5576-409: The material, its dipole moment per unit volume. If P is uniform, a macroscopic separation of charge is produced only at the surfaces where P enters and leaves the material. For non-uniform P , a charge is also produced in the bulk. Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of

5658-500: The network; whether unilateral or bilateral, active or passive, linear or non-linear. This law, also called Kirchhoff's second law , or Kirchhoff's loop rule , states the following: The directed sum of the potential differences (voltages) around any closed loop is zero. Similarly to Kirchhoff's current law, the voltage law can be stated as: ∑ i = 1 n V i = 0 {\displaystyle \sum _{i=1}^{n}V_{i}=0} Here, n

5740-409: The other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis . Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated. The equations introduce the electric field , E ,

5822-518: The physicist and mathematician James Clerk Maxwell , who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside . Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at

5904-404: The plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components is simply connected. Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with

5986-567: The relationship between an electric field and electric charges : an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space . Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles ; no north or south magnetic poles exist in isolation. Instead,

6068-420: The same equations expressed using tensor calculus or differential forms (see § Alternative formulations ). The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On

6150-499: The same physics, i.e. trajectories of charged particles, or work done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor : the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension. Such modified definitions are conventionally used with

6232-414: The same start and endpoint ( p ( 0 ) = q ( 0 ) {\displaystyle p(0)=q(0)} and p ( 1 ) = q ( 1 ) {\displaystyle p(1)=q(1)} ), then p {\displaystyle p} can be continuously deformed into q {\displaystyle q} while keeping both endpoints fixed. Explicitly, there exists

6314-402: The set of morphisms Hom Π ( X ) ⁡ ( x , y ) {\displaystyle \operatorname {Hom} _{\Pi (X)}(x,y)} in the fundamental groupoid of X {\displaystyle X} has only one element. In complex analysis : an open subset X ⊆ C {\displaystyle X\subseteq \mathbb {C} }

6396-436: The solution is { i 1 = 1 1100 A i 2 = 4 275 A i 3 = − 3 220 A {\displaystyle {\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\[6pt]i_{2}={\frac {4}{275}}{\text{A}}\\[6pt]i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}} The current i 3 has

6478-515: The sum of currents flowing into that node is equal to the sum of currents flowing out of that node; or equivalently: The algebraic sum of currents in a network of conductors meeting at a point is zero. Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be succinctly stated as: ∑ i = 1 n I i = 0 {\displaystyle \sum _{i=1}^{n}I_{i}=0} where n

6560-452: The surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility . A surface (two-dimensional topological manifold ) is simply connected if and only if it is connected and its genus (the number of handles of the surface) is 0. A universal cover of any (suitable) space X {\displaystyle X}

6642-434: The total, bound, and free charge and current density by ρ = ρ b + ρ f , J = J b + J f , {\displaystyle {\begin{aligned}\rho &=\rho _{\text{b}}+\rho _{\text{f}},\\\mathbf {J} &=\mathbf {J} _{\text{b}}+\mathbf {J} _{\text{f}},\end{aligned}}} and use

6724-539: Was found that the known values for ε 0 {\displaystyle \varepsilon _{0}} and μ 0 {\displaystyle \mu _{0}} give c ≈ 2.998 × 10 8   m/s {\displaystyle c\approx 2.998\times 10^{8}~{\text{m/s}}} , then already known to be the speed of light in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In

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