But shouldn't K be an integer? Is there is something wrong or that this is the idealization that we do to the solenoid? Wouldn't it be way off the correct value? Thank you in advance. BvU Science Advisor. Homework Helper.
Conductivity said:. BvU said:. No reason. If you have an 11 cm coil with 10 turns, you have Don't we idealize a solenoid as a number of circular coils? If dx has 2 turns then we multiply by 2, If it has k turns then we multiply by k. But a dx piece can't have a 2. You must log in or register to reply here.
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12.2: The Biot-Savart Law
Derive Ampere's law from Biot-Savart law?We have seen that mass produces a gravitational field and also interacts with that field.
Charge produces an electric field and also interacts with that field. Since moving charge that is, current interacts with a magnetic field, we might expect that it also creates that field—and it does. The equation used to calculate the magnetic field produced by a current is known as the Biot-Savart law. It is an empirical law named in honor of two scientists who investigated the interaction between a straight, current-carrying wire and a permanent magnet. This law enables us to calculate the magnitude and direction of the magnetic field produced by a current in a wire.
Since this is a vector integral, contributions from different current elements may not point in the same direction. Consequently, the integral is often difficult to evaluate, even for fairly simple geometries.
The following strategy may be helpful. A short wire of length 1. The rest of the wire is shielded so it does not add to the magnetic field produced by the wire. Calculate the magnetic field at point Pwhich is 1 meter from the wire in the x -direction. Since the current segment is much smaller than the distance xwe can drop the integral from the expression.
This approximation is only good if the length of the line segment is very small compared to the distance from the current element to the point. If not, the integral form of the Biot-Savart law must be used over the entire line segment to calculate the magnetic field. Calculate the magnetic field at the center of this arc at point P. We can determine the magnetic field at point P using the Biot-Savart law.
The radial and path length directions are always at a right angle, so the cross product turns into multiplication. Then we can pull all constants out of the integration and solve for the magnetic field.
As we integrate along the arc, all the contributions to the magnetic field are in the same direction out of the pageso we can work with the magnitude of the field.An electric current flowing in a conductor, or a moving electric charge, produces a magnetic field, or a region in the space around the conductor in which magnetic forces may be detected.
The value of the magnetic field at a point in the surrounding space may be considered the sum of all the contributions from each small element, or segment, of a current-carrying conductor. The Biot-Savart law states how the value of the magnetic field at a specific point in space from one short segment of current-carrying conductor depends on each factor that influences the field.
In the first place, the value of the magnetic field at a point is directly proportional to both the value of the current in the conductor and the length of the current-carrying segment under consideration. The value of the field depends also on the orientation of the particular point with respect to the segment of current. As this angle gets smaller, the field of the current segment diminishes, becoming zero when the point lies on a line of which the current element itself is a segment.
In addition, the magnetic field at a point depends upon how far the point is from the current element. At twice the distance, the magnetic field is four times smaller, or the value of the magnetic field is inversely proportional to the square of the distance from the current element that produces it.
Using the Biot-Savart Law requires calculus.
Those are infinitesimal magnetic field elements and wire elements. But we can use a simpler version of the law for a perfectly straight wire. If we straighten out the wire and do some calculus, the law comes out as muu-zero I divided by 2pir. Or in other words, the magnetic field, B, measured in teslas is equal to the permeability of free space, muu-zero, which is always 1.
So this equation helps us figure out the magnetic field at a radius r from a straight wire carrying a current I. The equation gives us the magnitude of the magnetic field, but a magnetic field is a vector, so what about the direction? The magnetic field created by a current-carrying wire takes the form of concentric circles.
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. There's already a question like this here so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question. Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law? I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.
In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow? Addendum : In mathematics and science it is important to keep in mind the distinction between the historical and the logical development of a subject. Knowing the history of a subject can be useful to get a sense of the personalities involved and sometimes to develop an intuition about the subject. The logical presentation of the subject is the way practitioners think about it.
It encapsulates the main ideas in the most complete and simple fashion.
From this standpoint, electromagnetism is the study of Maxwell's equations and the Lorentz force law. Everything else is secondary, including the Biot-Savart law. It may be true that in days of yore people measured the force resulting from a filamentary current, discovering the Biot-Savart law, and then in turn used that as inspiration to construct Maxwell's equations. If that's how it actually happened historically, fine.
But this is analogous to some alien archaeologist 10 million years from now finding a skeletal hand and foot in the Earth. From the hand, the archaeologist comes to understand what the animal who had that hand liked to do with it: that it could grasp and use tools and so on. From the foot, the archaeologist it comes to understand that the animal it belonged to walked on two legs and that it typically weighed in adulthood around pounds.
Only later does the archaeologist that the hand and the foot both belonged to the same animal--a human being. But the nature of the work means that the puzzle of what a human being was has to be broken down into chunks that can be individually understood before the whole picture can come together.
That said, it would be backwards to suggest the hand and the foot are more fundamental than the human being itself. The Maxwell equations have been constructed to be consistent with the Biot-Savart law and other pieces of information, like Coulomb's law. Thus, you can derive Biot-Savart from Maxwell, but not the other way around, for Maxwell is more general and all-encompassing.
If you already know the Lorentz force law, you can infer the strength of the magnetic field from a wire just by shooting charged test particles near the wire and observing their motion.
But this calls into question how you already know the Lorentz force law, and so on.It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot—Savart law is fundamental to magnetostaticsplaying a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot—Savart law should be replaced by Jefimenko's equations.
The Biot—Savart law is used for computing the resultant magnetic field B at position r in 3D-space generated by a flexible current I for example due to a wire. A steady or stationary current is a continual flow of charges which does not change with time and the charge neither accumulates nor depletes at any point. The law is a physical example of a line integralbeing evaluated over the path C in which the electric currents flow e.
The equation in SI units is . The symbols in boldface denote vector quantities. The integral is usually around a closed curvesince stationary electric currents can only flow around closed paths when they are bounded. However, the law also applies to infinitely long wires this concept was used in the definition of the SI unit of electric current—the Ampere —until 20 May Holding that point fixed, the line integral over the path of the electric current is calculated to find the total magnetic field at that point.
The application of this law implicitly relies on the superposition principle for magnetic fields, i. There is also a 2D version of the Biot-Savart equation, used when the sources are invariant in one direction. The resulting formula is:. The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire. If the conductor has some thickness, the proper formulation of the Biot—Savart law again in SI units is:.
In the case of a point charged particle q moving at a constant velocity vMaxwell's equations give the following expression for the electric field and magnetic field: . These equations were first derived by Oliver Heaviside in However, this language is misleading as the Biot—Savart law applies only to steady currents and a point charge moving in space does not constitute a steady current.
The Biot—Savart law can be used in the calculation of magnetic responses even at the atomic or molecular level, e. The Biot—Savart law is also used in aerodynamic theory to calculate the velocity induced by vortex lines. In the aerodynamic application, the roles of vorticity and current are reversed in comparison to the magnetic application.
In Maxwell's paper 'On Physical Lines of Force',  magnetic field strength H was directly equated with pure vorticity spinwhereas B was a weighted vorticity that was weighted for the density of the vortex sea.
Hence the relationship. The electric current equation can be viewed as a convective current of electric charge that involves linear motion.Gabriella Sciolla. Ampere's Law and its application to determine the magnetic field produced by a current; examples using a thick wire and a thick sheet of current.
John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Eric Hudson, Dr. Sen-Ben Liao. Back to Top. Introduction of the Biot-Savart Law for finding the magnetic field due to a current element in a current-carrying wire.
Worked example using the Biot-Savart Law to calculate the magnetic field due to a linear segment of a current-carrying wire or an infinite current-carrying wire. Uses Biot-Savart Law to determine the magnetic force between two parallel infinite current-carrying wires. Worked example using the Biot-Savart Law to calculate the magnetic field on the axis of a circular current loop.
Description and tabular summary of problem-solving strategy for the Biot-Savart Law, with a finite current segment and a circular current loop as examples.
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Description and tabular summary of problem-solving strategy for Ampere's Law, with an infinite wire, ideal solenoid, and ideal toroid as examples. Find the magnetic field everywhere due to a slab carrying a non-uniform current density. Solution is included after problem. Find the magnetic field everywhere due to the current distribution in a coaxial cable. Find the current through a hairpin-shaped wire loop to produce the given magnetic field at a symmetry point. A long current-carrying wire runs down the center of an ideal solenoid; find the magnetic force on the wire due to the solenoid and find the velocity of a particle inside the solenoid that doesn't feel the field of the wire.
Determine the magnetic field produced everywhere in space around a line segment carrying current. Determine the magnetic field at the center of an arc of current.
Determine the magnetic field at the center of a rectangle of current. Determine the magnetic field at the center of a hairpin of current. Determine the magnetic field along the axis between two infinite wires and determine where the field is the greatest.
Determine the magnetic field everywhere around a wire with a non-uniform current density. Find the magnetic field produced by two perpindicular rays of wire. Describe the application of Biot-Savart and Ampere's Laws; characterize magnetic attraction or repulsion between steady current configurations.
Use Ampere's Law to find the magnetic field due to an infinitely long current-carrying wire; then calculate a circulation involving eight infinite currents and discuss the utility of Ampere's Law. Find the magnetic field everywhere due to a long, hollow cylindrical conductor carrying a uniform current distribution. Find the magnetic field everywhere due to a uniform current distribution in a long cylindrical conductor with an off-center cylindrical hole.
Find the magnetic field at the center of a square configuration of four infinitely long current-carrying wires. Find the magnetic field of a standard solenoid and compare it to the magnetic field produced by a spinning cylinder with a uniform surface charge.
I think I have a proof for you though you may find it unsatisfying. For the most part, I'm following the proof on the wikipedia page you link to. I do avoid the dirac delta function however. Starting from the Biot-Savart law:. One of your criticisms of the proof was that the type of integral is left unspecified. The functions we need to integrate are Riemann integrable and thus Lebesgue integrable as wellso for the purposes of choosing particular proofs, I'm going to proceed based on the Riemann definition though this is an arbitrary choice on my part.
A problem we encounter already is that the Biot-Savart law is an improper integral. We solve this problem by saying the integral is the Cauchy principle value, which is defined in terms of a limit this is relevant.
Next, we're going to take the curl operator outside the integral. Since both the curl and integral are defined in terms of limits, this is equivalent to exchanging the order of limits, which is generally acceptable given certain convergence criteria e. There are a few different convergence theorems that may be appropriate. Applying curl to both sides of the equation, and the vector calculus identity for the curl of a curl:. Edit: As pointed out, if we were to use our definition of integral, the answer would just come out to 0.
Remember that exchanging limits is allowed given certain convergence criteria? If the integrals or integrands don't converge, then exchanging the limits is an error based on the formulations I've chosen. In this case, the integrand doesn't converge. We have shown however, that excluding the divergent point, the integral is 0. We've already shown that the integral over the first domain is 0.
Based on this:. The steps are largely the same as before, except all derivative operators are outside the integral. This leads to the following equation:. The only difference compared with our initial proof is that the Laplacian operator is outside the integral. The integral here can be evaluated, followed by the Laplacian operator. I hope I have been able to find a proof using Lebesgue integrals, where the differentiations under the integral signs are justified, if I am not wrong, by proved mathematical results.
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