Ampere's law - Top Future Point

Come friends, today we will know about Ampere’s law. In classical electromagnetism , Ampere’s law (not to be confused with Ampere’s law of force that André-Marie Ampere discovered in 1823) [1] relates the integrated magnetic field to the electric current through the loop around a closed loop passing. James Clerk Maxwell (not Ampere) obtained this using hydrodynamics in his 1861 published paper “On Physical Lines of Force” [2] In 1865 he generalized the equation to apply to time-varying currents by adding displacement current. Did.The term, resulting in the modern form of the law, sometimes called the Ampere –Maxwell law , is one of Maxwell’s equations that form the basis of classical electromagnetism.

Maxwell’s Basic Circuit Law

In 1820 the Danish physicist Hans Christian Oersted discovered that an electric current creates a magnetic field around it, when he observed that the needle of a compass next to a current-carrying wire twisted such that the needle was perpendicular to the wire. [6] [7] He investigated and discovered the laws that govern the vicinity of a direct current-carrying wire: [8]

Magnetic field lines surround a current carrying wire.
Magnetic field lines lie in a plane perpendicular to the wire.
If the direction of the current is reversed, the direction of the magnetic field is reversed.
The field strength is directly proportional to the magnitude of the current.
The field strength at any point is inversely proportional to the distance of the point from the wire.

This led to considerable research into the relationship between electricity and magnetism. Andre-Marie Ampere investigated the magnetic force between two current-carrying wires, discovering Ampere ‘s law of force . In the 1850s the Scottish mathematical physicist James Clerk Maxwell generalized these results and others into a single mathematical law. The original form of Maxwell’s circuital law, which he derived in 1855 in his paper “On Faraday’s Lines of Force” [9] on the basis of its analogy to hydrodynamics , describes magnetic fields as electric currents .relates to those that generate them. It determines the magnetic field associated with a given current or the current associated with a given magnetic field.

The basic circuit law for constant constant currents flowing in a closed circuit applies only to the magnetostatic state. For systems with electric fields that change over time, the basic law (as described in this section) must be modified to include a term known as Maxwell’s correction (see below).

Equivalent form

The basic circuit law can be written in several different forms, all of which are ultimately equivalent:

An “integral form” and an “differential form”. The forms are exactly equivalent, and are related to the Kelvin–Stokes theorem (see the ” Proofs ” section below).
Forms using SI units , and those using CGS units . Other units are possible, but rare. This section will use SI units, with CGS units to be discussed later.
Forms using B or H magnetic fieldsThese two forms use total current density and free current density respectively. The B and H fields are related by theunion equation: B = μ 0 H in non-magnetic materials where μ 0 is themagnetic constant.

Explanation

The integral of the basic circuit law is an integral of the magnetic field around some closed curve C (arbitrary but must be closed ) . The curve C in turn binds both a surface S , through which an electric current passes (again arbitrary but not closed – since no three-dimensional quantity is enclosed by S ), and encloses the current. The mathematical statement of the law is the relationship between the total amount of magnetic field around a path (the line integral) due to a current passing through that enclosed path (the surface integral).

In terms of total current, the line integral of the (which is the sum of both free current and bound current) magnetic B-field (in teslas, t) around the closed curve C is proportional to the total current I _ENC passing through a surface S ( attached by c ). ^{In terms of free current, the integral line of the magnetic H-field ( in amperes per meter, A m −1} ) around a closed curve C is equal to the free current I _{f, enc} passing through the surface S.

{\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\mathrm {enc} }} — Ampere’s law

\mathbf{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} — Ampere’s law

	integral form	differential form
B- Using field and total current	$\oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\mathrm {enc} }$	$\mathbf{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J}$
H- Using Field and Free Current	$\oint _{C}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\mathbf {J} _{\mathrm {f} }\cdot \mathrm {d} \mathbf {S} =I_{\mathrm {f,enc} }$	$\mathbf {\nabla } \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }$

J is the total current density ( in amperes per square meter , A m^-2 ),
J _f is simply the free current density,
C is the closed line integrated around the closed curve _C ,
_S denotes a 2 – D surface integration over S surrounded by C ,
The vector is the dot product,
d l is a minimal element (a differential) of the curve C ( that is, a vector whose magnitude is equal to the length of the minimal line element, and is the direction given by the tangent to the curve C )
d s is the vector area of a minimal surface element of a S (that is, a vector with magnitude equal to the area of the minimal surface element, and the direction normal to the surface s . The normal should correspond with the orientation of the direction c to the right From Hand’s rule), see below for further explanation of curve C and surface S.
× is the curl operator.

Ambiguities and Signature Conventions

There are many ambiguities in the above definitions which require explanation and choice of convention.

First, three of these terms are associated with sine ambiguity: the line integral C _can move around the loop in any direction (clockwise or counterclockwise); The vector field d S can point to either of the two directions normal to the surface ; And Ienc is the net current passing _through the surface S , which means that the current passing through one direction decreases the current in the other – but any direction can be chosen as positive. These ambiguities are resolved by the right-hand rule: the palm of the right hand—pointing toward the area of integration, and the index finger—pointing in the direction of integration, with the outstretched thumb pointing in the direction that should be chosen for the vector field. d for S, Also , the current passing in the same direction as d S should be considered positive. The right hand grip rule can also be used to determine the signal.
Second, there are infinitely many possible surfaces S that have curve C as their boundary. (Imagine a soap film on a wire loop, which can be deformed by twisting the wire). Which of these surfaces to choose? For example, if the loops are not in the same plane, there is no one obvious choice. The answer is that it doesn’t matter; By Stokes’ theorem, the integral is the same for any surface with limit C since the integrand is a smooth sphere (ie the curl of the exact). In practice, one usually chooses the most convenient surface (with a given range) to integrate with.

Free current vs bound current

Electric current generated in the simplest textbook situations would be classified as “free current”—for example, current that passes through a wire or battery. In contrast, “bound current” arises in the context of bulk materials that can be magnetized and/or polarized. (All ingredients can to some extent.)

When a material is magnetized (for example, by placing it in an external magnetic field), electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction , by which a subtle current is generated. When the currents of all these atoms are put together, they produce an effect similar to that of a macroscopic current, which continuously moves around the magnetic object. This magnetization current J _m is a contribution to the “bound current”. Another source of bound current is bound charge. When an electric field is applied, atoms in the positive and negative bound charges can separate over distances in the polarizable material, and when the bound charges move, the polarization changes, creating another contribution to the “bound current”, the polarization current. J _P.

The total current density J due to free and bound charges is then:

\mathbf {J} =\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {M} }+\mathbf {J} _{\mathrm {P} }\,,

With J _f “free” or “conduction” current density.

All currents are subtly fundamentally the same. Nevertheless, there are often practical reasons to treat bound current differently from free current. For example, bound current usually arises from atomic dimensions, and one may wish to take advantage of a simpler principle for larger dimensions. The result is that the more subtle Ampere’s circuit law, expressed as B and the microcurrent (which includes the free, magnetizing and polarizing currents), is sometimes referred to as H and below only in terms of free current. are kept in the same manner. For a detailed definition of free current and bound current, and proof that the two formulations are equivalent, see the “Proof” section below.

Drawbacks of the Basic Formulation of the Circuit Law

There are two important issues with respect to the Circle Act that require close scrutiny. First, there is a problem regarding the continuity equation for electric charge. In vector calculus, the identity of Curl’s divergence states that the Curl’s divergence of a vector field must always be zero. That’s why

\nabla \cdot (\nabla \times \mathbf {B} )=0\,,

And so the original Ampere’s circuit law implies that

\nabla \cdot \mathbf {J} =0\,.

But in general, reality follows the continuity equation for electric charge:

\nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}\,,

which is non-zero for the time-varying charge density. An example occurs in a capacitor circuit where time-varying charge densities exist on the plates.

Second, there is an issue regarding the propagation of electromagnetic waves. For example, in the blank space, where

\mathbf {J} =\mathbf {0} \,.

The circuit law means that

\nabla \times \mathbf {B} =\mathbf {0} \,,

But for the electric charge to maintain continuity with the continuity equation, we must have

\nabla \times \mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\,.

To remedy these situations, the contribution of the displacement current must be added to the current term in the circuit law.

James Clerk Maxwell conceived of the displacement current as a polarizing current in a dielectric vortex ocean, which he used to model the magnetic field hydrodynamically and mechanically. ^[17] He added this displacement current to Ampere’s circuital law in his 1861 paper “On Physical Lines of Force” on Equation 112. ^[18]

Displacement current

In empty space, the displacement current is related to the rate at which the electric field changes.

The above contribution to the displacement current in a dielectric is also present, but a major contribution to the displacement current is related to the polarization of individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, charges in molecules can move slightly under the influence of an electric field. The positive and negative charges in the molecules separate under the applied field, leading to an increase in the polarization state, which is expressed as the polarization density P. The changing state of polarization is equivalent to a current.

The two contributions to the displacement current are combined by defining the displacement current as:

\mathbf {J} _{\mathrm {D} }={\frac {\partial }{\partial t}}\mathbf {D} (\mathbf {r} ,\,t)\,,

where the electric displacement field is defined as:

\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} =\varepsilon _{0}\varepsilon _{\mathrm {r} }\mathbf {E} \,,

where 0 is the power constant, r is the relative permittivity constant, and _p isthe polarization density. Substituting this form for D in the displacement current expression , it has two components:

\mathbf {J} _{\mathrm {D} }=\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\partial \mathbf {P} }{\partial t}}\,.

The first term on the right is present everywhere, even in vacuum. It does not involve any actual movement of charge, but still has an associated magnetic field, as if it were a real current. Some authors apply the name displacement current only to this contribution . ^[19]

The second term on the right is displacement current, originally conceived by Maxwell, to be associated with the polarization of individual molecules of a dielectric material.

Maxwell’s original explanation for the displacement current focused on the condition occurring in the dielectric media. In the modern post-aether era, the concept has been extended to apply to situations in which no physical media are present, for example, to the vacuum between the plates of a charging vacuum capacitor. Displacement current is appropriate today because it meets several requirements of electromagnetic theory: accurate prediction of magnetic fields in regions where there is no free flow; prediction of wave propagation of electromagnetic fields; and conservation of electric charge in cases where the charge density is time-varying. See displacement stream for more discussion.

Extension of the Basic Law: Ampere-Maxwell Equation

Subsequently, the circuit equation is extended by including the polarization current, thereby treating the limited applicability of the original circuit law.

In the case of treating free charges separately from bound charges, including Maxwell’s correction of the equation H -field ( H -field is used, because it includes magnetization currents, then J _m , does not appear clearly see H -field and also note ):

\oint _{C}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\left(\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S}

(integral form), where H is the magnetic H field (also called “auxiliary magnetic field”, “intensity of the magnetic field”, or just “magnetic field”), D is the electric displacement field, and J is the conduction current attached to _f or free current density. differentially,

\mathbf {\nabla } \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\,.

On the other hand, assuming all charges to be on the same basis (whether bound or free charges), the generalized Ampere’s equation, also known as the Maxwell–Ampere equation, is in integral form (see the “Proofs” section below):

\oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\left(\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S}

differentially,

\mathbf {\nabla } \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}

Both forms include magnetization current densities in J^[21] as well as conduction and polarization current densities. That is, the current density on the right side of the Ampere-Maxwell equation is:

\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {D} }+\mathbf {J} _{\mathrm {M} }=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {P} }+\mathbf {J} _{\mathrm {M} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,,

where the current density J is the _displacement current , and J is the current density contribution in fact due to the movement of charges, both free and bound. Because d = , the charge continuity issue with Ampere ‘s original construction is no longer a problem. ^[22]Because of this word in ₀e _/t _Now wave propagation in free space is possible.

With the addition of the displacement current, Maxwell was able to hypothesize (correctly) that light was a form of electromagnetic wave. See Electromagnetic wave equations for a discussion of this important discovery.

Proof of equivalence

Proof that the formulations of the circuit law with respect to the free current are equivalent to the sums with the total current.
In this proof, we will show that the equation

\nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}

is equal to the equation

{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,.

Note that we are dealing only with differential forms, not integral forms, but this is sufficient because by the Kelvin–Stokes theorem the differential and integral forms are equivalent in each case.

We introduce the polarization density P , which has the following relation to E and D :

\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \,.

Next, we introduce the magnetization density M , which has the following relation to B and H :

{\frac {1}{\mu _{0}}}\mathbf {B} =\mathbf {H} +\mathbf {M}

and the following relation to the bound current:

{\begin{aligned}\mathbf {J} _{\mathrm {bound} }&=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\\&=\mathbf {J} _{\mathrm {M} }+\mathbf {J} _{\mathrm {P} },\end{aligned}}

Where from

\mathbf {J} _{\mathrm {M} }=\nabla \times \mathbf {M} ,

is called the magnetization current density, and

\mathbf {J} _{\mathrm {P} }={\frac {\partial \mathbf {P} }{\partial t}},

is the polarization current density. Taking the equation for B :

{\begin{aligned}{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )&=\mathbf {\nabla } \times \left(\mathbf {H} +\mathbf {M} \right)\\&=\mathbf {\nabla } \times \mathbf {H} +\mathbf {J} _{\mathrm {M} }\\&=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {P} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {J} _{\mathrm {M} }.\end{aligned}}

Consequently, referring to the definition of bound current:

{\begin{aligned}{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )&=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {bound} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\\&=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}},\end{aligned}}

as it was to be shown.

Ampere’s circuit law in CGS units

Come on friends, now we will know about Ampere’s circuital law in CGS units. In cgs units, the integral form of the equation, including Maxwell’s correction, reads