Biot Savart Law

12 Key excerpts on "Biot Savart Law"

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  Electromagnetism

  • I. S. Grant, W. R. Phillips(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Equation (1.10) for the electrostatic field of a distribution of charges, and gives the magnetic field as the sum of contributions from the infinite number of elementary lengths which comprise the circuits. The Biot–Savart law was deduced from measurements on the fields due to currents in wires made by Ampère, Biot and Savart in the years around 1820. Ampère’s law can be deduced from the Biot–Savart law and can thus be regarded as distilled from experimental observations in the same way as Gauss’ law was the result of observations on the forces between charges.

  Figure 4.15 . The field at the point P is made up of contributions dB from elements like the one shown at Q.

  We will introduce the Biot–Savart law by using it to determine the field due to a current in a complete circuit s, part of which is the straight section shown in Figure 4.15 . The field at the point P (position vector r with respect to the origin O) can be considered to be made up of contributions dB from elements of length dl′ like the one at the point Q. The magnitude of the contribution from each element is proportional to the current I in the wire, and inversely proportional to the square of the distance of the element from the point P. Let the position vector of the point Q be r′. The direction of the contribution dB from the element at Q is perpendicular to the vector dl′, which is in the direction of the current in the wire at the point Q, and perpendicular to the vector (r – r′) which is in the direction of the line joining Q to P. Combining all the properties listed above we can write the contribution dB as

  The total field at the point P is obtained by integrating around the complete circuit s ,

  (4.32)

  This is the Biot–Savart law .

  Let us work out the field at a point P when the circuit has a long straight section. If the distance of the point P from the wire is small compared to the length of the straight portion and the circuit is closed far from P the field can be considered with little error to be that arising from a current in an infinitely long straight wire. It is simplest to choose the origin of coordinates where the perpendicular from P hits the wire, as shown in Figure 4.16 . Let the distance OP be r , and the distance OQ be z . The variable r′ in Equation (4.32) now becomes ez z , where ez is a unit vector in the direction of the current in the wire, and the variable dl′ becomes ez dz . The vector dl′ × (r – r′) has magnitude QP dz sin θ , where θ

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  University Physics Volume 2

  • William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
  • 2016(Publication Date)
  • Openstax

    (Publisher)

  The following strategy may be helpful. Problem-Solving Strategy: Solving Biot-Savart Problems To solve Biot-Savart law problems, the following steps are helpful: 1. Identify that the Biot-Savart law is the chosen method to solve the given problem. If there is symmetry in the problem comparing B → and d l → , Ampère’s law may be the preferred method to solve the question. 2. Draw the current element length d l → and the unit vector r ^ , noting that d l → points in the direction of the current and r ^ points from the current element toward the point where the field is desired. 3. Calculate the cross product d l → × r ^ . The resultant vector gives the direction of the magnetic field according to the Biot-Savart law. 4. Use Equation 12.4 and substitute all given quantities into the expression to solve for the magnetic field. Note all variables that remain constant over the entire length of the wire may be factored out of the integration. 5. Use the right-hand rule to verify the direction of the magnetic field produced from the current or to write down the direction of the magnetic field if only the magnitude was solved for in the previous part. Example 12.1 Calculating Magnetic Fields of Short Current Segments A short wire of length 1.0 cm carries a current of 2.0 A in the vertical direction (Figure 12.3). 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 P, which is 1 meter from the wire in the x-direction. Chapter 12 | Sources of Magnetic Fields 537 12.1 Figure 12.3 A small line segment carries a current I in the vertical direction. What is the magnetic field at a distance x from the segment? Strategy We can determine the magnetic field at point P using the Biot-Savart law. Since the current segment is much smaller than the distance x, we can drop the integral from the expression. The integration is converted back into a summation, but only for small dl, which we now write as Δl.

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  Fundamentals of Electrical Engineering, Part 1

  • S. B. Lal Seksena, Kaustuv Dasgupta(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The first method is the mechanical means of magnetization. One end of a permanent magnet is swiped over the surface of the substance repetitively. Each time some of the tiny magnets get aligned. After a sufficient time of action a significant number of tiny magnets aligned along with the direction of the magnetic field of external magnet. In the central region of the substance all the S poles and N poles will face each other annulling their magnetic effects. But at one end there will be free N poles while at another end there will be free S poles. (see Fig. 5.1b) If we send some direct current through a coil wounded over the substance there will be magnetic field associated with the current flow. This will act upon the tiny magnets and align them the same way. The polarity of the magnet will depend on the direction of the current flow and the sense of winding. This is known as the electro-magnet. If we send more and more current through the coil more and more numbers of tiny magnets will get aligned resulting the increase of the magnetic strength. After a certain level of current almost all the tiny magnets will be organized and the strength of the electromagnet will reach its saturation. There after there will be no effect of increment of the current on the magnetic strength. 5.4 Biot-Savart’s Law In 1820, two French scientists Jean-Bapiste Biot and Felix Savart discovered a relationship between marring charges, i.e. current and the magnetic field produced by it. Later this has been named after then as ‘Biot-Savart’s Law’. The law establishes a relationship between a current carrying conductor and the magnetic field due to its current flow. Let us assume a current carrying conductor is carrying a current I. We consider an elementary length — Δ l. The agentive field at a point P due to current flowing — Δ l is — Δ B. While the relative position vector between P and — Δ l is r – and vector is aclined angle. The law states the following: a. ⏐ — Δ B ⏐∝ ⏐ — Δ l ⏐ ……….

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  Einstein's Theory of Relativity

  • Max Born(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

         Up to the early nineteenth century, electricity and magnetism were regarded as two regions of phenomena which were similar in some respects but quite separate and independent. A bridge was eagerly sought between the two regions, but for a long time without success. At last Oersted (1820) discovered that the magnetic needle is deflected by voltaic currents. In the same year Biot and Savart discovered the quantitative law of this phenomenon, which Laplace formulated in terms of action at a distance. This law is very important for us, for the reason that in it there occurs a constant, peculiar to electromagnetism and of the nature of a velocity, which showed itself later to be identical with the velocity of light.

         Biot and Savart established that the current flowing in a straight wire neither attracts nor repels a magnetic pole, but strives to drive it around in a circle about the wire (Fig. 83 ), so that the positive pole moves in the sense of a right-handed screw turned from below (contrary to the hands of a watch) about the (positive) direction of the current. The quantitative law can be brought into the simplest form by supposing the conducting wire to be divided into a number of short pieces of length / and writing down the effect of these current elements, from which the effect of the whole current is obtained by summation. We shall state the law of a current element only for the special case in which the magnetic pole lies in the plane that passes through the middle part of the element and is perpendicular to its direction (Fig. 84 ). Then the force that acts on the magnet pole of unit strength, i.e., the magnetic intensity of field H in this plane, is perpendicular to the line connecting the pole with the midpoint of the current element, and is directly proportional to the current intensity J and to its length l, and inversely proportional to the square of the distance r:

         Fig. 83 The magnetic field H surrounding a current J.

         Fig. 84

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  Classical Electrodynamics

  • John David Jackson(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Steady-state magnetic phenomena are characterized by no change in the net charge density anywhere in space. Consequently in magnetostatics ∇ i J = 0 (5.3) We now proceed to discuss the experimental connection between current and magnetic-flux density and to establish the basic laws of magnetostatics. 5.2 | BIOT AND SAVART LAW In 1819 Oersted observed that wires carrying electric currents produced deflections of permanent magnetic dipoles placed in their neighborhood. Thus the currents were sources of magnetic-flux density. Biot and Savart (1820), first, and Ampère (1820–1825), in much more elaborate and thorough experiments, established the basic experimental laws relating the magnetic induction B to the currents and established the law of force between one current and another. Although not in the form in which Ampère deduced it, the basic relation is the following. If dl is an element of length (pointing in the direction of current flow) of a filamentary wire that carries a current I and x is the coordinate vector from the element of length to an observation point P, as shown in Fig. 5.1, then the elemental flux density d B at the point P is given in magnitude and direction by Chapter 5 Magnetostatics, Faraday’s Law, Quasi-Static Fields 180 d kI d B l x x = ( ) | | × 3 (5.4) It should be noted that (5.4) is an inverse square law, just as is Coulomb’s law of electrostatics. However, the vector character is very different. A word of caution about (5.4). There is a temptation to think of (5.4) as the magnetic equivalent of the electric field (1.3) of a point charge and to identify I d l as the analog of q. Strictly speaking this is incorrect. Equation (5.4) has meaning only as one element of a sum over a continuous set, the sum representing the magnetic induction of a current loop or circuit.

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  Electromagnetic Field Theory Fundamentals

  • Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  A major breakthrough came in 1820 when Hans Christian Oersted discovered experimentally that a magnetic needle was deflected by a current in a wire. This event bridged the gap between the science of electricity and magnetism. Scientists immediately realized that electric currents are also sources of magnetic fields. Within a short time after Oersted’s discovery, Biot and Savart experi-mentally formulated an equation to determine the magnetic flux density 177 178 5 Magnetostatics Figure 5.1 Magnetic lines of flux surrounding a bar magnet at a point produced by a current-carrying conductor. We now view the Biot–Savart law as the magnetic equivalent of Coulomb’s law. By 1825 Andr´ e Marie Amp` ere had discovered the existence of magnetic force between current-carrying conductors and formulated a set of qualitative relationships based upon a series of experiments. These discoveries lead to the development of electric machines we use in our daily lives. This chapter is devoted to the study of magnetostatics ; i.e., the mag-netic fields produced by steady currents. We begin our discussion with the Biot-Savart law and use it as a basic tool to calculate the magnetic field set up by any given distribution of currents. 5.2 The Biot–Savart law ................................. It has been found experimentally that the magnetic flux density produced at a point P from an element of length d of a filamentary wire carrying a steady current I , as shown in Figure 5.2, is d B = k Id × a R R 2 In this equation, d B is the elemental magnetic flux density in teslas (T), where one tesla is equal to one weber per square meter (Wb/m 2 ), d is an element of length in the direction of the current, a R is the unit vector pointing from d to point P , the point P is at a distance R from the current element d , and k is the constant of proportionality. Figure 5.2 Magnetic flux density at point P produced by current element at Q

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  Principles of Electrodynamics

  • Melvin Schwartz(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  As a rule we will find the techniques of this section to be of marginal utility because it will normally be no harder to calculate the field directly than it will be to go through the vector potential. Furthermore, there is no situation analogous to a set of conductors with fixed potentials to determine our boundary conditions. Nevertheless, when all else fails, this method may provide a path to the solution we seek.

  4-3 THE BIOT-SAVART LAW

  Making use of our expression for A(r), Eq. (3-4-14) , we can proceed directly to find a general expression for B(r).

  This result is called the Biot-Savart law. It is a simple prescription for finding the magnetic field B at any point r by integrating over the current distribution.

  We often deal with situations where the actual current distribution within a wire itself is not important and can be averaged over. This situation occurs when the thickness of the wire is small compared with our distance from it. In these cases we can integrate over the cross-sectional area of the wire and replace the volume integral in Eq. (4-3-1) by a line integral along the wire. We have then

  It is very tempting at this point to say that the little bit of wire dl′ gives rise to a magnetic field dB(r) given by

  If we then used Eq. (3-5-17) to find the force exerted by dB on a charge q at position r and moving with velocity V, we would find

  The problem, as can be plainly seen, is that this force violates Newton’s third law. It is not directed along a line joining the current element dl′ with the charge q.

  It is not possible for us to completely resolve this dilemma at this point in our course. We can however point out one serious problem. We are not entitled to write down Eq. (4-3-3)

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  Introduction to Electrodynamics

  • David J. Griffiths(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  5.1.1. I 1 I 2 d (1) (2) FIGURE 5.20 5.2 The Biot-Savart Law 227 Example 5.6. Find the magnetic field a distance z above the center of a circular loop of radius R, which carries a steady current I (Fig. 5.21). B θ θ d B d l′ z R r FIGURE 5.21 Solution The field d B attributable to the segment d l  points as shown. As we integrate d l  around the loop, d B sweeps out a cone. The horizontal components cancel, and the vertical components combine, to give B(z ) = μ 0 4π I  dl  r 2 cos θ. (Notice that d l  and r are perpendicular, in this case; the factor of cos θ projects out the vertical component.) Now, cos θ and r 2 are constants, and  dl  is simply the circumference, 2π R, so B(z ) = μ 0 I 4π  cos θ r 2  2π R = μ 0 I 2 R 2 ( R 2 + z 2 ) 3/2 . (5.41) For surface and volume currents, the Biot-Savart law becomes B(r) = μ 0 4π  K(r  ) × ˆ r r 2 da  and B(r) = μ 0 4π  J(r  ) × ˆ r r 2 d τ  , (5.42) respectively. You might be tempted to write down the corresponding formula for a moving point charge, using the “dictionary” (Eq. 5.30): B(r) = μ 0 4π q v × ˆ r r 2 , (5.43) 228 Chapter 5 Magnetostatics but this is simply wrong. 11 As I mentioned earlier, a point charge does not con- stitute a steady current, and the Biot-Savart law, which only holds for steady cur- rents, does not correctly determine its field. The superposition principle applies to magnetic fields just as it does to electric fields: if you have a collection of source currents, the net field is the (vector) sum of the fields due to each of them taken separately. Problem 5.8 (a) Find the magnetic field at the center of a square loop, which carries a steady current I . Let R be the distance from center to side (Fig. 5.22). (b) Find the field at the center of a regular n-sided polygon, carrying a steady cur- rent I . Again, let R be the distance from the center to any side. (c) Check that your formula reduces to the field at the center of a circular loop, in the limit n → ∞.

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  Inductance

  Loop and Partial

  • Clayton R. Paul(Author)
  • 2011(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  J involves only the coordinates of the location of the source current. You can verify in spherical coordinates (see the Appendix) that

  (2.50)

  where aR is a unit vector pointing from the current to the field point. The vector identity in (2.49) is, of course, sensible since it is the vector counterpart to the scalar result using the chain rule. Therefore, we obtain

  (2.51)

  which is the Biot-Savart law for determining B given in (2.11) . If the current forms a closed loop (as all dc currents must), ∇ · A = 0, but it is not necessary to define the divergence of A to be zero in order to obtain (2.47) .

  The solutions in (2.47) apply to any coordinate system. If we specialize them to a rectangular coordinate system, we obtain

  (2.52a)

  This is a significant result because it says that (1) each component of J produces the corresponding component of A, and (2) the direction of the resulting

  Ax ,Ay , Az

  is the same as the direction of the corresponding

  Jx , Jy , Jz

  that produced it! In other words, a current that is directed solely in the z direction will produce a vector magnetic potential that is solely in the z direction parallel to the Jz at all points in the space around the current! For a current distributed over a surface s, these results become

  (2.52b)

  If the current is a line current whose contour is l such as in a wire, these become

  (2.52c)

  Note that unlike the Biot-Savart law, which is an inverse-square law where B depends on the square of the inverse distance between the current and the B field, the magnetic vector potential simply depends on the inverse of the distance R between the current and the component of A that it produces. In some problems it is simpler to determine the components of A from (2.47) , which for rectangular coordinates are given in (2.52) , and then simply determine B by computing its curl mechanically from B = ∇ × A in the appropriate coordinate system, than it is to compute B

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  Classical Electromagnetism in a Nutshell

  • Anupam Garg(Author)
  • 2012(Publication Date)
  • Princeton University Press

    (Publisher)

  a , which can then be approximated by a line of dipoles to get the far field. Check your answer by directly superposing the fields of the separate wires. Sketch the field.

  FIGURE 4.9. Force between two current loops.

  Exercise 22.4 Find the magnetic field in the plane of a circular current loop, as a function of the distance from the center. The Biot-Savart line integral is best done numerically, even though an analytic expression for it exists in terms of elliptic integrals. Plot your results. What is the area in which the field is uniform to 5%? to 10%?

  Exercise 22.5 Repeat the previous exercise for a square current loop. The integral can be done in simple closed form this time. Plot the field along a line from the center to the midpoint of a side, and to a corner.

  The Biot-Savart law immediately enables us to find the forces between two current loops, carrying currents I 1 and I 2 (see fig. 4.9 ). We find the field at loop 2 due to loop 1 and use eq. (21.8) to calculate the force on an element of loop 2. Adding together all length elements gives the net force on loop 2,

  with r 21 = r 2 − r 1 being the vector from the element

  dℓ 1

  to

  dℓ

  2 . Expanding out the triple vector product, we obtain

  The first term can be written as When integrated over loop 2, it yields zero. Hence,

  By interchanging the indices 1 and 2, and noting that r 12 = −r 21 , we see that

  In other words, the forces on the loops are equal and opposite, and obey Newton’s third law.

  It is useful at this point to discuss more closely the meaning of the term magnetostatics and what we mean by a steady current. After all, the charges in a current-carrying wire are not static. Indeed, if the loop is curved, as it must be to be closed, the individual charges carrying the current undergo acceleration. If we remember that accelerating charges produce radiation, the question arises as to how the magnetic field can be constant in time. The answer is that the radiative effects cancel out when we consider a large number of charges that flow in a continuous stream. The stipulation of large numbers is important. If the discreteness of the individual charges is evident, it is meaningless to talk of a steady current. Many authors prefer the term magnetism of steady currents

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  Geophysical Field Theory and Method, Part A

  Gravitational, Electric, and Magnetic Fields

  • Alexander A. Kaufman(Author)
  • 1992(Publication Date)
  • Academic Press

    (Publisher)

  We are again faced with the problem of the closed circle. Therefore, the Biot -Savart law cannot be used to calculate the magnetic field, and instead of it we have to formulate a boundary-value problem. In this connection it is useful to consider the system of equations for the field H. In accordance with Eqs. (IV.273) we obtain curlH = 0 CurlH = 0 div u.H = r u« div P, Div]LH = -]La Div P r (IV.293) As in the previous case the field H has a source origin only, and the distribution of its sources is defined by Eqs. (IV.275)-(IV.278). Of course, one type of fictitious sources depends on the field H, and this fact requires the formulation of a boundary-value problem too. Thus, determination of the magnetic field can be, in principle, aCCOm-plished in two ways. One of them is based on the solution of the system (IV.291), while the other allows us to find the field H and then, making use of Eq, (IY.258), to determine B. Taking into account the fact that H is a source field, the second approach in general is more preferable since it permits us to introduce a scalar potential U, which essentially simplifies the calculation of the field. Case 5 Remanent Magnetization and Conduction Currents are Absent Suppose that a magnetic substance is placed in an external magnetic field B o ' which is known. Then, due to the induced magnetization, molecular currents arise that generate the secondary magnetic field B s ' Therefore, the total magnetic field B consists of two parts. Since conduction currents and remanent magnetization are absent, the 510 IV Magnetic Fields system of field equations is markedly simplified and we have B curl-= 0 f.L n x (B Z -~) = 0 f.Lz f.LI divB = 0 (IV.294) Consequently, the volume and surface densities of molecular currents are 1 j = -(grad f.L X B), f.Lf.Lo (IV.295) and they can be determined, provided that the field B is known. This study shows that in general even at regular points the field has a vortex origin.

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  General Physics. Electromagnetism Optics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2023(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  By exploiting the superposition principle, the magnetic field generated by the whole circuit is Laplace’s law  B = d  B γ ∫ = μ 0 4π ids r 2  u t ×  u r γ ∫ , where the integral is extended to the whole circuit γ in which a current i flows. A current is due to the motion, with drift velocity  v, of N elementary charges contained in a portion ds of a circuit, thus a current element, supposing that charge carriers are posi- tive, can be written as i d  s =  j idA  u t ( ) d  s = nevdAds  u t = ndVev  u t = Nev  u t and the magnetic field generated in point P at distance r, from a beam of N moving charges is therefore  B = μ 0 4π Nev r 2  u t ×  u r . Hence, even a single charge generates in the space an electric and a magnetic field, both being time dependent, since, in general, r and v are time dependent, given by  B = μ 0 4π ev r 2  u t ×  u r  E = 1 4πε 0 e r 2  u r ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ so in the case of non relativistic velocities, the relationship  B = μ 0 ε 0  v ×  E = 1 c 2  v ×  E holds. The constant c = 1 ε 0 μ 0 exactly matches the speed of light in the vacuum. The previous relationship implies that electric fields and magnetic fields are correlated, i.e. they have not a separate existence: it can be therefore presumed that only electromag- netic fields exist and that static electric fields and static magnetic fields are just peculiar manifestations of these more general fields. Laplace’s elementary law cannot be verified experimentally, since a current element i d  s cannot be isolated from the rest of the circuit. Moreover, the superposition principle assumed by Laplace in order to deduce the formula which allows the calculation of a mag- netic field was a hypothesis which was experimentally verified only later. It is therefore a mathematical expression whose validity is assumed only because it gives correct results for finite size circuits. Chapter 10 Laws of Magnetostatics 155

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