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This result is independent of the size and shape of the closed curve enclosing a current. Let I be the current flowing in the direction as shown in Fig.

The magnetic field is produced around the conductor. The magnetic lines of force are concentric circles in the XY plane as shown by dotted lines. Let the magnitude of the magnetic field induction produced at a point P at distance r from the conductor is. Consider a close circular loop as shown in figure. Therefore, angle between them is zero. A long straight wire of radius R carries a steady current I that is uniformly distributed through the cross-section of the wire.

For finding the behavior of magnetic field due to this wire, let us divide the whole region into two parts. Because the total current passing through the plane of the circle is I. Because the current is uniform over the cross-section of the wire. Save my name, email, and website in this browser for the next time I comment. By eSaral. February 14, Class 12 - Physics.

Let the magnitude of the magnetic field induction produced at a point P at distance r from the conductor is Consider a close circular loop as shown in figure.

Leave a comment Cancel Please enter comment. Please enter your name. Search Anything. Complete Chapter Notes. Subjective Questions. Revision Series. Video Lectures. Test Series. Ask Doubts. Previous Yr. Topicwise Qs with Sol. Class 10 Study Material. Recent Posts. There is some Error.The original form of Maxwell's circuital law, which he derived as early as in his paper "On Faraday's Lines of Force" [3] based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them.

It determines the magnetic field associated with a given current, or the current associated with a given magnetic field. The original circuital law is only a correct law of physics in a magnetostatic situation, where the system is static except possibly for continuous steady currents within closed loops.

For systems with electric fields that change over time, the original law as given in this section must be modified to include a term known as Maxwell's correction see below. The original circuital law can be written in several different forms, which are all ultimately equivalent:.

The integral form of the original circuital law is a line integral of the magnetic field around some closed curve C arbitrary but must be closed.

The curve C in turn bounds both a surface S which the electric current passes through again arbitrary but not closedâ€”since no three-dimensional volume is enclosed by Sand encloses the current. The mathematical statement of the law is a relation between the total amount of magnetic field around some path line integral due to the current which passes through that enclosed path surface integral.

In terms of total currentwhich is the sum of both free current and bound current the line integral of the magnetic B -field in teslasT around closed curve C is proportional to the total current I enc passing through a surface S enclosed by C.

There are a number of ambiguities in the above definitions that require clarification and a choice of convention. The electric current that arises in the simplest textbook situations would be classified as "free current"â€”for example, the current that passes through a wire or battery.

All materials can to some extent. When a material is magnetized for example, by placing it in an external magnetic fieldthe electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction, creating a microscopic current. When the currents from all these atoms are put together, they create the same effect as a macroscopic current, circulating perpetually around the magnetized object.

This magnetization current J M is one contribution to "bound current". The other source of bound current is bound charge. When an electric field is applied, the positive and negative bound charges can separate over atomic distances in polarizable materialsand 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:. All current is fundamentally the same, microscopically. Nevertheless, there are often practical reasons for wanting to treat bound current differently from free current. For example, the bound current usually originates over atomic dimensions, and one may wish to take advantage of a simpler theory intended for larger dimensions.It is now one of Maxwell's equationswhich form the basis of classical electromagnetism.

The law relates magnetic fields to electric currents that produce them. A scientist can use Ampere's law to determine the magnetic field associated with a given current or current associated with a given magnetic field, if there is no time changing electric field present.

The law can be written in two forms, the "integral form" and the "differential form".

**AmpĂ¨re's Law: Crash Course Physics #33**

The forms are equivalent, and related by the Kelvinâ€”Stokes theorem. It can also be written in terms of either the B or H magnetic fields. Again, the two forms are equivalent see the " proof " section below.

In all other cases the law is incorrect unless Maxwell's correction is included see below. See below for further explanation of the curve C and surface S.

There are a number of ambiguities in the above definitions that require clarification and a choice of convention. These ambiguities are resolved by the right-hand rule : With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area d S.

Also the current passing in the same direction as d S must be counted as positive. The right hand grip rule can also be used to determine the signs. Second, there are infinitely many possible surfaces S that have the curve C as their border.

Imagine a soap film on a wire loop, which can be deformed by moving the wire. Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter; it can be proven that any surface with boundary C can be chosen. By the Kelvinâ€”Stokes theoremthis equation can also be written in a "differential form".

Again, this equation only applies in the case where the electric field is constant in time; see below for the more general form. In SI units, the equation states:. The electric current that arises in the simplest textbook situations would be classified as "free current.

All materials can to some extent.The original form of Maxwell's circuital law, which he derived as early as in his paper "On Faraday's Lines of Force" [3] based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them.

## Ampereâ€™s law and its applications in daily life

It determines the magnetic field associated with a given current, or the current associated with a given magnetic field. The original circuital law is only a correct law of physics in a magnetostatic situation, where the system is static except possibly for continuous steady currents within closed loops. For systems with electric fields that change over time, the original law as given in this section must be modified to include a term known as Maxwell's correction see below.

The original circuital law can be written in several different forms, which are all ultimately equivalent:. The integral form of the original circuital law is a line integral of the magnetic field around some closed curve C arbitrary but must be closed. The curve C in turn bounds both a surface S which the electric current passes through again arbitrary but not closedâ€”since no three-dimensional volume is enclosed by Sand encloses the current.

The mathematical statement of the law is a relation between the total amount of magnetic field around some path line integral due to the current which passes through that enclosed path surface integral. In terms of total currentwhich is the sum of both free current and bound current the line integral of the magnetic B -field in teslasT around closed curve C is proportional to the total current I enc passing through a surface S enclosed by C.

There are a number of ambiguities in the above definitions that require clarification and a choice of convention. The electric current that arises in the simplest textbook situations would be classified as "free current"â€”for example, the current that passes through a wire or battery. All materials can to some extent. When a material is magnetized for example, by placing it in an external magnetic fieldthe electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction, creating a microscopic current.

When the currents from all these atoms are put together, they create the same effect as a macroscopic current, circulating perpetually around the magnetized object. This magnetization current J M is one contribution to "bound current".

The other source of bound current is bound charge. When an electric field is applied, the positive and negative bound charges can separate over atomic distances in polarizable materialsand 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:. All current is fundamentally the same, microscopically.

Nevertheless, there are often practical reasons for wanting to treat bound current differently from free current. For example, the bound current usually originates over atomic dimensions, and one may wish to take advantage of a simpler theory intended for larger dimensions. For a detailed definition of free current and bound current, and the proof that the two formulations are equivalent, see the " proof " section below.

There are two important issues regarding the circuital law that require closer scrutiny. First, there is an issue regarding the continuity equation for electrical charge. In vector calculus, the identity for the divergence of a curl states that the divergence of the curl of a vector field must always be zero. But in general, reality follows the continuity equation for electric charge :. 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 free spacewhere. To treat these situations, the contribution of displacement current must be added to the current term in the circuital law.Maxwell's Equations Home. The 4th Maxwell's Equation On this page, we'll explain the meaning of the last of Maxwell's Equations, Ampere's Lawwhich is given in Equation [1]: [Equation 1] Ampere was a scientist experimenting with forces on wires carrying electric current.

He was doing these experiments back in the s, about the same time that Farday was working on Faraday's Law. Ampere and Farday didn't know that there work would be unified by Maxwell himself, about 4 decades later. Forces on wires aren't particularly interesting to me, as I've never had occassion to use the very complicated equations in the course of my work which includes a Ph.

So, I'm going to start by presenting Ampere's Law, which relates a electric current flowing and a magnetic field wrapping around it: [Equation 2] Equation [2] can be explained: Suppose you have a conductor wire carrying a current, I. Then this current produces a Magnetic Field which circles the wire. The left side of Equation [2] means: If you take any imaginary path that encircles the wire, and you add up the Magnetic Field at each point along that path, then it will numerically equal the amount of current that is encircled by this path which is why we write for encircled or enclosed current.

Let's do an example for fun. Suppose we have a long wire carrying a constant electric current, I [Amps]. What is the magnetic field around the wire, for any distance r [meters] from the wire?

Let's look at the diagram in Figure 1. We have a long wire carrying a current of I Amps. We want to know what the Magnetic Field is at a distance r from the wire.

## Ampere's Law

So we draw an imaginary path around the wire, which is the dotted blue line on the right in Figure 1: Figure 1. Ampere's Law [Equation 2] states that if we add up integrate the Magnetic Field along this blue path, then numerically this should be equal to the enclosed current I. Now, due to symmetry, the magnetic field will be uniform not varying at a distance r from the wire.

The path length of the blue path in Figure 1 is equal to the circumference of a circle of radius r :.

### AmpĂ¨re's circuital law

If we are adding up a constant value for the magnetic field we'll call it Hthen the left side of Equation [2] becomes simple: [Equation 3] Hence, we have figured out what the magnitude of the H field is. And since r was arbitrary, we know what the H-field is everywhere. However, the H field is a Vector Fieldwhich means at every location is has both a magnitude and a direction.

The direction of the H-field is everywhere tangential to the imaginary loops, as shown in Figure 2. The right hand rule determines the sense of direction of the magnetic field: Figure 2.

We can rewrite Ampere's Law in Equation [2]: [Equation 4] On the right side equality in Equation [4], we have used Stokes' Theorem to change a line integral around a closed loop into the curl of the same field through the surface enclosed by the loop S.

We can also rewrite the total current as the surface integral of the Current Density J : [Equation 5] So now we have the original Ampere's Law Equation [2] rewritten in terms of surface integrals Equations [4] and [5].

Hence, we can substitute them together and get a new form for Ampere's Law: [Equation 6] Now, we have a new form of Ampere's Law: the curl of the magnetic field is equal to the Electric Current Density. If you are an astute learner, you may notice that Equation [6] is not the final form, which is written in Equation [1].

There is a problem with Equation [6], but it wasn't until the s that James Clerk Maxwell figured out the problem, and unified electromagnetics with Maxwell's Equations. So let's look at what is wrong with it.

First, I have to throw out another vector identity - the divergence of the curl of any vector field is always zero: [Equation 7] So let's take the divergence of Ampere's Law as written in Equation [6]: [Equation 8] So Equation [8] follows from Equations [6] and [7].

But it says that the divergence of the current density J is always zero. Is this true? If the divergence of J is always zero, this means that the electric current flowing into any region is always equal to the electric current flowing out of the region no divergence.

This seems somewhat reasonable, as electric current in circuits flows in a loop. But let's look what happens if we put a capacitor in the circuit: Figure 3. A Voltage Applied to A Capacitor. Now, we know from electric circuit theory that if the voltage is not constant for example, any periodic wave, such as the 60 Hz voltage that comes out of your power outlets then current will flow through the capacitor.

That is, we have I not equal to zero in Figure 3.It is now one of Maxwell's equationswhich form the basis of classical electromagnetism. The law relates magnetic fields to electric currents that produce them. A scientist can use Ampere's law to determine the magnetic field associated with a given current or current associated with a given magnetic field, if there is no time changing electric field present.

The law can be written in two forms, the "integral form" and the "differential form". The forms are equivalent, and related by the Kelvinâ€”Stokes theorem. It can also be written in terms of either the B or H magnetic fields. Again, the two forms are equivalent see the " proof " section below. In all other cases the law is incorrect unless Maxwell's correction is included see below. See below for further explanation of the curve C and surface S. There are a number of ambiguities in the above definitions that require clarification and a choice of convention.

First, three of these terms are associated with sign ambiguities: the line integral could go around the loop in either direction clockwise or counterclockwise ; the vector area d S could point in either of the two directions normal to the surface; and I enc is the net current passing through the surface Smeaning the current passing through in one direction, minus the current in the other directionâ€”but either direction could be chosen as positive. These ambiguities are resolved by the right-hand rule : With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area d S.

Also the current passing in the same direction as d S must be counted as positive. The right hand grip rule can also be used to determine the signs. Second, there are infinitely many possible surfaces S that have the curve C as their border. Imagine a soap film on a wire loop, which can be deformed by moving the wire. Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter; it can be proven that any surface with boundary C can be chosen.

By the Kelvinâ€”Stokes theorem, this equation can also be written in a "differential form". Again, this equation only applies in the case where the electric field is constant in time; see below for the more general form. In SI units, the equation states:. The electric current that arises in the simplest textbook situations would be classified as "free current.The physical origin of this force is that each wire generates a magnetic field, following the Biotâ€”Savart lawand the other wire experiences a magnetic force as a consequence, following the Lorentz force law.

This is a good approximation if one wire is sufficiently longer than the other, so that it can be approximated as infinitely long, and if the distance between the wires is small compared to their lengths so that the one infinite-wire approximation holdsbut large compared to their diameters so that they may also be approximated as infinitely thin lines. The value of k A depends upon the system of units chosen, and the value of k A decides how large the unit of current will be.

In the SI system, [1] [2]. The general formulation of the magnetic force for arbitrary geometries is based on iterated line integrals and combines the Biotâ€”Savart law and Lorentz force in one equation as shown below.

To determine the force between wires in a material medium, the magnetic constant is replaced by the actual permeability of the medium. For the case of two separate closed wires, the law can be rewritten in the following equivalent way by expanding the vector triple product and applying Stokes' theorem: [8]. In this form, it is immediately obvious that the force on wire 1 due to wire 2 is equal and opposite the force on wire 2 due to wire 1, in accordance with Newton's 3rd law.

Through differentiation, it can be shown that:. As Maxwell noted, terms can be added to this expression, which are derivatives of a function Q r and, when integrated, cancel each other out. Thus, Maxwell gave "the most general form consistent with the experimental facts" for the force on ds arising from the action of ds': [11]. Q is a function of r, according to Maxwell, which "cannot be determined, without assumptions of some kind, from experiments in which the active current forms a closed circuit.

Ritz left k undetermined in his theory. If wire 1 is also infinite, the integral diverges, because the total attractive force between two infinite parallel wires is infinity. In fact, what we really want to know is the attractive force per unit length of wire 1. Then the force vector felt by wire 1 is:. As expected, the force that the wire feels is proportional to its length. The force per unit length is:.

The direction of the force is along the y-axis, representing wire 1 getting pulled towards wire 2 if the currents are parallel, as expected. From Wikipedia, the free encyclopedia. Redirected from Ampere's circuital theorem. Electrical network. Covariant formulation. Electromagnetic tensor stressâ€”energy tensor. Serway's principles of physics: a calculus based text Fourth ed. Monk Physical chemistry: understanding our chemical world.

New York: Chichester: Wiley. Archived from the original on 20 August Retrieved 8 August Chow Introduction to electromagnetic theory: a modern perspective. Boston: Jones and Bartlett. American Journal of Physics. Bibcode : AmJPh.

Electromagnetic Theory.

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