Physics
Magnetic Field of a Current-Carrying Wire
The magnetic field of a current-carrying wire refers to the region around the wire where a magnetic force is exerted. When an electric current flows through a wire, it generates a magnetic field around the wire according to the right-hand rule. The strength and direction of the magnetic field depend on the magnitude and direction of the current.
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11 Key excerpts on "Magnetic Field of a Current-Carrying Wire"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
Magnetic field lines form in concentric circles around a cylindrical current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the right hand grip rule. The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength decreases inversely proportional to the distance.) ________________________ WORLD TECHNOLOGIES ________________________ Solenoid Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely-spaced loops to form a coil or solenoid enhances this effect. A device so formed around an iron core may act as an electromagnet , generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet except that its strength and polarity are determined by the current flowing through the coil. The magnetic field generated by a steady current I (a constant flow of electric charges in which charge is neither accumulating nor depleting at any point) is described by the Biot–Savart law : where the integral sums over the wire length where vector d ℓ is the direction of the current, μ 0 is the magnetic constant, r is the distance between the location of d ℓ and the location at which the magnetic field is being calculated, and r^ is a unit vector in the direction of r . A slightly more general way of relating the current I to the B-field is through Ampère's law : ________________________ WORLD TECHNOLOGIES ________________________ where the line integral is over any arbitrary loop and I enc is the current enclosed by that loop.
eBook - ePubElectromagnetics Explained
A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
- Ron Schmitt(Author)
- 2002(Publication Date)
- Newnes(Publisher)
In fact, all magnetic fields are generated indirectly by moving electric charges. It is a fundamental fact of nature that moving electrons, as well as any other charges, produce a magnetic field when in motion. Electrical currents in wires also produce magnetic fields because a current is basically the collective movement of a large number of electrons. A steady (DC) current through a wire produces a magnetic field that encircles the wire, as shown in Figure 3.1. Figure 3.1 Magnetic field lines surrounding a current-carrying wire. A single charge moving at constant velocity also produces a tubular magnetic field that encircles the charge, as shown in Figure 3.2. However, the field of a single charge decays along the axis of propagation, with the maximum field occurring in the neighborhood of the charge. The law that describes the field is called the Biot-Savart law, named after the two French scientists who discovered it. Figure 3.2 Magnetic field lines surround a moving electron. It is interesting to note that if you were to move along at the same velocity as the charge, the magnetic field would disappear. In that frame of reference, the charge is stationary, producing only an electric field. Therefore, the magnetic field is a relative quantity. This odd situation hints at the deep relationship between Einstein’s relativity and electromagnetics, which you will learn about in detail in Chapter 6. The magnetic field direction, clockwise or counterclockwise, depends on which direction the current flows. You can use the “right hand rule” for determining the magnetic field direction. Using Figure 3.3 as a guide, extend your hand flat and point your thumb in the direction of the current (i.e., current is defined as the flow of positive charge, which is opposite to the flow of electrons). Now curl the rest of your fingers to form a semicircle
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
All moving charged particles produce magnetic fields. Moving point charges, such as electrons, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles. Magnetic field lines form in concentric circles around a cylindrical current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the right hand grip rule (see figure). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength decreases inversely proportional to the distance.) Solenoid ________________________ WORLD TECHNOLOGIES ________________________ Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely-spaced loops to form a coil or solenoid enhances this effect. A device so formed around an iron core may act as an electromagnet , generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet except that its strength and polarity are determined by the current flowing through the coil. The magnetic field generated by a steady current I (a constant flow of electric charges in which charge is neither accumulating nor depleting at any point) is described by the Biot–Savart law : where the integral sums over the wire length where vector d ℓ is the direction of the current, μ 0 is the magnetic constant, r is the distance between the location of d ℓ and the location at which the magnetic field is being calculated, and r ̂ is a unit vector in the direction of r .
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Key Ideas ● The magnetic field set up by a current-carrying con- ductor can be found from the Biot–Savart law. This law asserts that the contribution d B → to the field produced by a current-length element i d s → at a point P located a distance r from the current element is d B → = μ 0 ___ 4π id s → × r ̂ _______ r 2 (Biot–Savart law). Here r ̂ is a unit vector that points from the element toward P. The quantity μ 0 , called the permeability con- stant, has the value 4π × 10 −7 T · m/A ≈ 1.26 × 10 −6 T · m/A. ● For a long straight wire carrying a current i, the Biot– Savart law gives, for the magnitude of the magnetic field at a perpendicular distance R from the wire, B = μ 0 i ____ 2πR (long straight wire). ● The magnitude of the magnetic field at the center of a circular arc, of radius R and central angle ϕ (in radians), carrying current i, is B = μ 0 iϕ _____ 4πR (at center of circular arc). C H A P T E R 2 9 What Is Physics? One basic observation of physics is that a moving charged particle produces a magnetic field around itself. Thus a current of moving charged particles produces a magnetic field around the current. This feature of electromagnetism, which is the combined study of electric and magnetic effects, came as a surprise to the people who discovered it. Surprise or not, this feature has become enormously 887 29.1 MAGNETIC FIELD DUE TO A CURRENT important in everyday life because it is the basis of countless electromagnetic devices. For example, a magnetic field is produced in maglev trains and other devices used to lift heavy loads. Our first step in this chapter is to find the magnetic field due to the current in a very small section of current-carrying wire. Then we shall find the mag- netic field due to the entire wire for several different arrangements of the wire. Calculating the Magnetic Field Due to a Current Figure 29.1.1 shows a wire of arbitrary shape carrying a current i. We want to find the magnetic field B → at a nearby point P.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
C H A P T E R 2 9 Magnetic Fields Due to Currents 29-1 MAGNETIC FIELD DUE TO A CURRENT Learning Objectives After reading this module, you should be able to . . . 726 What Is Physics? One basic observation of physics is that a moving charged particle produces a mag- netic field around itself. Thus a current of moving charged particles produces a mag- netic field around the current. This feature of electromagnetism, which is the combined study of electric and magnetic effects, came as a surprise to the people who discov- ered it. Surprise or not, this feature has become enormously important in everyday life because it is the basis of countless electromagnetic devices. For example, a mag- netic field is produced in maglev trains and other devices used to lift heavy loads. Our first step in this chapter is to find the magnetic field due to the current in a very small section of current-carrying wire. Then we shall find the magnetic field due to the entire wire for several different arrangements of the wire. 29.01 Sketch a current-length element in a wire and indi- cate the direction of the magnetic field that it sets up at a given point near the wire. 29.02 For a given point near a wire and a given current- length element in the wire, determine the magnitude and direction of the magnetic field due to that element. 29.03 Identify the magnitude of the magnetic field set up by a current-length element at a point in line with the direction of that element. 29.04 For a point to one side of a long straight wire carrying current, apply the relationship between the magnetic field magnitude, the current, and the distance to the point. 29.05 For a point to one side of a long straight wire car- rying current, use a right-hand rule to determine the direction of the field vector. 29.06 Identify that around a long straight wire carrying current, the magnetic field lines form circles.
eBook - ePub- I. S. Grant, W. R. Phillips(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
The field is cylindrically symmetrical, and its magnitude is inversely proportional to the perpendicular distance from the wire. The direction of the field at any point is perpendicular to both the wire and the perpendicular from the point to the wire. The field lines are circles, and the direction of the arrows on the circles is given by the right-hand screw rule. This rule says that the circles go round in the direction the threads of a right-hand screw would rotate if it were screwed in the direction of the current. Figure 4.8 shows the magnetic field lines from a small loop of wire carrying a current. (The leads carrying the current into and out of the loop are close together so that their magnetic effects cancel. They are not drawn.) The field from the small current loop is axially symmetrical, i.e. it is independent of the azimuthal angle with respect to the axis of the coil. This means that a map of the field lines would look the same for any plane containing the axis of the coil. An important result from such field mapping exercises is that at distances from the loop large compared with the dimensions of the loop, the magnetic field varies in the same way as the electric field of an electric dipole. There is one obvious feature of the field lines on Figures 4.7 and 4.8 that is shared by the magnetic field lines arising from any current distribution. The lines of the field B are continuous. Unlike electrostatic field lines, lines of the field B have no beginning and no end. This means that there are no free ‘magnetic charges’ or ‘magnetic poles’, a fact which has important consequences for all of electromagnetism. Let us write this simple experimental fact in the form of a convenient mathematical equation. Consider a closed surface S around a volume V. Since the lines of the field B are continuous, as many lines enter the volume V as leave it. This means that the total outward flux of the field B over the surface S is zero, Figure 4.7
eBook - ePubIntroduction to Electromagnetism
From Coulomb to Maxwell
- Martin J N Sibley(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
Figure 3.3b shows why this is so.) In addition, there will be an equal and opposite force on the north pole due to the field surrounding the wire.FIGURE 3.3 (a) Magnetic field produced by a current-carrying wire and (b) plan view of wire/magnetic field.Let us try to find the magnetic field strength, δH 1 , at the north pole, due to the current element formed by I and dl . As we have just discussed, the current element produces a force on the north pole, and the north pole will produce an equal and opposite force on the current element. If we can find these two forces, and then equate them, we should get an expression for the magnetic field strength generated by the wire.Let us initially consider the field at dl due to the imaginary north pole of strength p N . As this north pole is a point source, it emits magnetic flux in a radial direction. Thus, we can write the flux density asDirect experimental measurement shows that the force on a current-carrying conductor placed in a magnetic field is given by(3.8)B N=rp N4 πr 2F = B I l(3.9)where B is the flux density of the magnetic field in which the wire is placed, I is the current flowing through the wire and l is the length of the wire. (We can intuitively reason that this equation is correct by noting that powerful electric motors require a large electric current and contain a large amount of wire – they are very heavy!)By combining Equations (3.8) and (3.9), we find that the force on the element dl due to the field emitted by the north pole isd F =(3.10)I d lp N4 πr 2Let us now turn our attention to the force on the north pole produced by the current element. The current element formed by the current I and the length dl produces a magnetic field strength of dH 1
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
When a current I exists in a coil of wire with N turns, each of area A, in the presence of a magnetic field of magni- tude B, the coil experiences a net torque that has a magnitude given by Equation 21.4, where is the angle between the direction of the magnetic field and the normal to the plane of the coil. The quantity NIA is known as the magnetic moment of the coil. τ = NIAB sin ϕ (21.4) 21.7 Magnetic Fields Produced by Currents An electric current produces a magnetic field, with different current geometries giving rise to different field patterns. For an infinitely long, straight wire, the mag- netic field lines are circles centered on the wire, and their direction is given by Right-Hand Rule No. 2 (see below). The magnitude of the mag- netic field at a radial distance r from the wire is given by Equation 21.5, where I is the current in the wire and 0 is a constant known as the permeability of free space ( μ 0 = 4π × 10 −7 T · m/A). B = μ 0 I _ 2πr (21.5) Focus on Concepts 683 Right-Hand Rule No. 2: Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the con- ventional current I, and the tips of the fingers will point in the direc- tion of the magnetic field → B . The magnitude of the magnetic field at the center of a flat circu- lar loop consisting of N turns, each of radius R and carrying a current I, is given by Equation 21.6. B = N μ 0 I _ 2R (21.6) The loop has associated with it a north pole on one side and a south pole on the other side. The side of the loop that behaves like a north pole can be predicted by using Right-Hand Rule No. 3: Curl the fingers of the right hand along the direction of the conven- tional current I around the loop or solenoid, and the thumb will point in the direction of the magnetic field at the center of the loop, or everywhere inside the solenoid, and toward their respec- tive north poles.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Figure 12.5 shows a section of an infinitely long, straight wire that carries a current I. What is the magnetic field at a point P, located a distance R from the wire? Figure 12.5 A section of a thin, straight current-carrying wire. The independent variable θ has the limits θ 1 and θ 2 . Let’s begin by considering the magnetic field due to the current element I d x → located at the position x. Using the right- hand rule 1 from the previous chapter, d x → × r ^ points out of the page for any element along the wire. At point P, therefore, the magnetic fields due to all current elements have the same direction. This means that we can calculate the net field there by evaluating the scalar sum of the contributions of the elements. With | d x → × r ^ | = (dx)(1)sin θ, we have from the Biot-Savart law (12.5) B = µ 0 4π ⌠ ⌡ wire I sin θdx r 2 . The wire is symmetrical about point O, so we can set the limits of the integration from zero to infinity and double the answer, rather than integrate from negative infinity to positive infinity. Based on the picture and geometry, we can write expressions for r and sin θ in terms of x and R, namely: 540 Chapter 12 | Sources of Magnetic Fields This OpenStax book is available for free at http://cnx.org/content/col12074/1.3 r = x 2 + R 2 sin θ = R x 2 + R 2 . Substituting these expressions into Equation 12.5, the magnetic field integration becomes (12.6) B = µ o I 2π ⌠ ⌡ 0 ∞ R dx (x 2 + R 2 ) 3/2 . Evaluating the integral yields (12.7) B = µ o I 2πR ⎡ ⎣ ⎢ x (x 2 + R 2 ) 1/2 ⎤ ⎦ ⎥ 0 ∞ . Substituting the limits gives us the solution (12.8) B = µ o I 2πR . The magnetic field lines of the infinite wire are circular and centered at the wire (Figure 12.6), and they are identical in every plane perpendicular to the wire. Since the field decreases with distance from the wire, the spacing of the field lines must increase correspondingly with distance.
eBook - ePub- Michael M. Mansfield, Colm O'Sullivan(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
19 Interactions between magnetic fields and electric currents; magnetic materials AIMS to understand the forces experienced by electric currents in magnetic fields to show how such forces may be utilised in instrument design to study the forces on charges which are moving in magnetic fields to investigate the nature of magnetic materials 19.1 Forces between currents and magnets Consider the situation shown in Figure 19.1. The current I in the wire produces a magnetic field at the location occupied by a pole of strength p. The contribution to the force on p due to the current I flowing in the element is directed perpendicularly to the page inwards and, from Equation (18.14), is given by From Newton's third law, there will be an equal and opposite force on the element due to the influence of the pole p, which is given by (19.1) and is directed perpendicularly to the page outwards. Let us now change the origin relative to which r is defined so that it is located at the pole p rather than at the element, that is in Equation (19.1). The force on the element due to the pole p is then given by Figure 19.1 The force exerted on the current element by the magnetic pole p is directed perpendicularly to the page outwards if p is an N pole. Figure 19.2 The force on the current element due to a magnetic field of flux density B at the element is given by Equation (19.2). Figure 19.3 The force on a straight length l of wire in a uniform magnetic field of flux density B is IlB sin θ, where I is the current in the wire and θ is the angle the wire makes with the direction of the magnetic field. Now the term in this equation is the magnetic flux density B at the element due to the pole p (recall Equation (18.15)) and hence we may write (19.2) This result describes equally well the force on a current element due to any source of magnetic field since, in the limit, the flux density at the element will be the same as that due to an equivalent single pole
eBook - PDF- Yung-Kuo Lim(Author)
- 1993(Publication Date)
- WSPC(Publisher)
PART 2 MAGNETOSTATIC FIELD AND ELECTROMAGNETIC FIELD QUASI-STATIONARY 1. MAGNETIC FIELD OF CURRENTS (2001-2038) 2001 A cylindrical wire of permeability p carries a steady current I. If the radius of the wire is R, find B and H inside and outside the wire. ( Wisconsin) Solution: Use cylindrical coordinates with the z-axis along the of the wire and the positive direction along the current flow, as shown in Fig. 2.1. On account of the uniformity of the current the current density is 8 L Fig. 2.1 Consider a point at distance r from the axis of the wire. Ampbre’s circuital law A H . & = I , where L is a circle of radius r with centre on the x-axis, gives for r > R, I 2rr H(r) = -ee , or POI 2rt B(r) = -ee since by symmetry H(r) and B(r) are independent of 8. For r < R, 147 148 Problcmr €9 Solrtionr on Elcciromagnciirm and the circuital law gives 2002 A long non-magnetic cylindrical conductor with inner radius a and outer radius b carries a current 1. The current density in the conductor is uniform. Find the magnetic field set up by this current as a function of radius (a) inside the hollow space ( r < a); (b) within the conductor (a < r < b); (c) outside the conductor (r > b). ( Wisconsin) Solution: in the conductor is Use cylindrical coordinates as in Problem 2001. The current density I = r(b2 - a2) ‘ The current passing through a cross-section enclosed by a circle of radius r, where a < r < b, is By symmetry, Ampkre’s circuital law gives (a) B = 0, (r < a). (b) B(r) = g. H e @ , (a < r < b ) . (c) B(r) = g e e , (r > b ) . 2003 The direction of the magnetic field of a long straight wire carrying 8 (a) in the direction of the current current is: Magnetortaiic Field and Quari-Staiionorp Elcctromagnciic Field 149 (b) radially outward (c) along lines circling the current Solution: The answer is (c). 2004 What is the magnetic field due to a long cable carrying 30,000 amperes (a) 3 x at a distance of 1 meter? Tesla, (b) 6 x Tesla, (c) 0.6 Tesla.
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