Physics
Curl of the Magnetic Field
The curl of the magnetic field is a measure of how the magnetic field changes from one point to another within a given region. It describes the rotation or circulation of the magnetic field around a point and is a fundamental concept in electromagnetism. The curl of the magnetic field is important in understanding the behavior of magnetic fields in various physical systems.
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8 Key excerpts on "Curl of the Magnetic Field"
eBook - PDFCalculus
Multivariable
- Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The coordinate definition of curl gives curl = ( (0) − ) + ( − (0) ) + ( − ) = 0 . (b) This vector field appears to be rotating around the -axis. By the right-hand rule, the circulation density around is negative, so we expect the -component of the curl to point down. The coordinate definition gives curl = ( (0) − (−) ) + ( − (0) ) + ( (−) − ) = −2 . (c) At first glance, you might expect this vector field to have zero curl, as all the vectors are parallel to the -axis. However, if you find the circulation around the curve in Figure 20.6, the sides contribute nothing (they are perpendicular to the vector field), the bottom contributes a negative quantity (the curve is in the opposite direction to the vector field), and the top contributes a larger positive quantity (the curve is in the same direction as the vector field and the magnitude of the vector field is larger at the top than at the bottom). Thus, the circulation around is positive and hence we expect the curl to be nonzero and point up. The coordinate definition gives curl = ( (0) − (0) ) + ( (−( + 1)) − (0) ) + ( (0) − (−( + 1)) ) = . Another way to see that the curl is nonzero in this case is to imagine the vector field representing the velocity of moving water. A boat sitting in the water tends to rotate, as the water moves faster on one side than the other. 20.1 THE CURL OF A VECTOR FIELD 1059 Figure 20.6: Rectangular curve in -plane Figure 20.7: Rotating flywheel Example 3 A flywheel is rotating with angular velocity and the velocity of a point with position vector is given by = × .
eBook - PDFCalculus
Multivariable
- William G. McCallum, Deborah Hughes-Hallett, Andrew M. Gleason, David O. Lomen, David Lovelock, Jeff Tecosky-Feldman, Thomas W. Tucker, Daniel E. Flath, Joseph Thrash, Karen R. Rhea, Andrew Pasquale, Sheldon P. Gordon, Douglas Quinney, Patti Frazer Lock(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
The induced magnetic field B (x, y, z ) is B (x, y, z ) = 2I c −y i + x j x 2 + y 2 , where c is the speed of light. In Example 5 on page 1074 we showed that curl B = 0 . (a) Compute the circulation of B around the circle C 1 in the xy-plane of radius a, centred at the origin, and oriented counterclockwise when viewed from above. (b) Use part (a) and Stokes’ Theorem to compute C2 B · d r , where C 2 is the ellipse x 2 + 9y 2 = 9 in the plane z = 2, oriented counterclockwise when viewed from above. Solution (a) On the circle C 1 , we have ‖ B ‖ = 2I/(ca). Since B is tangent to C 1 everywhere and points in the forward direction around C 1 , C1 B · d r = ‖ B ‖ · Length of C 1 = 2I ca · 2πa = 4πI c . (b) Let S be the conical surface extending from C 1 to C 2 in Figure 20.15. The boundary of this surface has two pieces, −C 2 and C 1 . The orientation of C 1 leads to the outward normal on S, which forces us to choose the clockwise orientation on C 2 . By Stokes’ Theorem, S curl B · d A = -C2 B · d r + C1 B · d r = − C2 B · d r + C1 B · d r . Since curl B = 0 , we have S curl B · d A = 0, so the two line integrals must be equal: C2 B · d r = C1 B · d r = 4πI c . 20.2 STOKES’ THEOREM 1081 y x z C1 −C2 n S Figure 20.15: Surface joining C1 to C2, oriented to satisfy the conditions of Stokes’ Theorem S1 n1 C S2 n2 Figure 20.16: The flux of a curl is the same through the two surfaces S1 and S2 if they determine the same orientation on the boundary, C Curl Fields A vector field F is called a curl field if F = curl G for some vector field G . Recall that if F = gradf , then f is called a potential function. By analogy, if a vector field F = curl G , then G is called a vector potential for F . The following example shows that the flux of a curl field through a surface depends only on the boundary of the surface.
eBook - PDFCalculus
Multivariable
- William G. McCallum, Deborah Hughes-Hallett, Daniel E. Flath, Andrew M. Gleason, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Ayse Arzu Sahin, Adam H. Spiegler, Jeff Tecosky-Feldman, Thomas W. Tucker, Aaron(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Contents 20.1 The Curl of a Vector Field . . . . . . . . . . . . . . . . 1000 Circulation Density . . . . . . . . . . . . . . . . . . . . . . 1000 Definition of the Curl . . . . . . . . . . . . . . . . . . . . 1001 Why Do the Two Definitions of Curl Give the Same Result? . . . . . . . . . . . . . . . . . . . 1004 Curl-Free Vector Fields . . . . . . . . . . . . . . . . . . . 1004 20.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1008 The Boundary of a Surface . . . . . . . . . . . . . . . . 1008 Calculating the Circulation from the Circulation Density . . . . . . . . . . . . . . . . . . . . . 1008 Curl-Free Vector Fields . . . . . . . . . . . . . . . . . . . 1010 Curl Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 20.3 The Three Fundamental Theorems . . . . . . . . . 1015 The Gradient and the Curl . . . . . . . . . . . . . . . . . 1015 The Curl and the Divergence . . . . . . . . . . . . . . 1016 Chapter Twenty THE CURL AND STOKES’ THEOREM 1000 Chapter 20 THE CURL AND STOKES’ THEOREM 20.1 THE CURL OF A VECTOR FIELD The divergence is a scalar derivative which measures the outflow of a vector field per unit volume. Now we introduce a vector derivative, the curl, which measures the circulation of a vector field. Imagine holding the paddle-wheel in Figure 20.1 in the flow shown by Figure 20.2. The speed at which the paddle-wheel spins measures the strength of circulation. Notice that the angular velocity depends on the direction in which the stick is pointing. If the stick is pointing horizontally the paddle- wheel does not spin; if the stick is vertical, the paddle wheel spins. Figure 20.1: A device for measuring circulation Figure 20.2: A vector field (in the planes = 1, = 2, = 3) with circulation about the -axis Circulation Density We measure the strength of the circulation using a closed curve.
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 (see figure at right). 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 electro-magnet , 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, with its strength and polarity 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 : where the line integral is over any arbitrary loop and I enc is the current enclosed by that loop.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
B μ Magnetic Field B → A magnetic field B → is defined in terms of the force F → B acting on a test particle with charge q moving through the field with velocity v → : F → B = q v → × B → . (28.1.2) The SI unit for B → is the tesla (T): 1 T = 1 N/(A · m) = 10 4 gauss. The Hall Effect When a conducting strip carrying a current i is placed in a uniform magnetic field B → , some charge carriers (with charge e) build up on one side of the conductor, creating a Review & Summary potential difference V across the strip. The polarities of the sides indicate the sign of the charge carriers. A Charged Particle Circulating in a Magnetic Field A charged particle with mass m and charge magnitude |q| mov- ing with velocity v → perpendicular to a uniform magnetic field B → will travel in a circle. Applying Newton’s second law to the cir- cular motion yields |q|vB = mv 2 ____ r , (28.4.2) Additional examples, video, and practice available at WileyPLUS 877 QUESTIONS from which we find the radius r of the circle to be r = mv ____ |q|B . (28.4.3) The frequency of revolution f, the angular frequency ω, and the period of the motion T are given by f = ω ___ 2π = 1 __ T = |q|B ____ 2πm . (28.4.6, 28.4.5, 28.4.4) Magnetic Force on a Current-Carrying Wire A straight wire carrying a current i in a uniform magnetic field experiences a sideways force F → B = iL → × B → . (28.6.2) The force acting on a current element i d L → in a magnetic field is d F → B = i d L → × B → . (28.6.4) The direction of the length vector L → or d L → is that of the current i. Torque on a Current-Carrying Coil A coil (of area A and N turns, carrying current i) in a uniform magnetic field B → will experience a torque τ → given by τ → = μ → × B → . (28.8.3) Here μ → is the magnetic dipole moment of the coil, with magni- tude μ = NiA and direction given by the right-hand rule. Orientation Energy of a Magnetic Dipole The orienta- tion energy of a magnetic dipole in a magnetic field is U(θ) = − μ → ⋅ B → .
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.
- Ozgur Ergul(Author)
- 2021(Publication Date)
- Wiley(Publisher)
Mathematically, this can be written ¯ B (¯ r 0 ) = [ ¯ ∇ × ¯ A (¯ r )] ¯ r =¯ r 0 , (4.114) i.e., the specific evaluation of the magnetic flux density at a position ¯ r 0 can be done by inserting ¯ r = ¯ r 0 after the curl operation is evaluated. 34 Given that the electric current distribution is 34 This is similar to ¯ E (¯ r 0 ) = − [ ¯ ∇ Φ(¯ r )] ¯ r =¯ r 0 . When the magnetic vector potential is found for a particular case (a specific position), the variance information (curl) may be lost. steady, the magnetic flux density can be found via the Biot-Savart law as ¯ B (¯ r ) = μ 0 J 0 b 4 π 2 π 0 b a ˆ a φ × (ˆ a z z − ˆ a ρ ρ ) ρ [( ρ ) 2 + z 2 ] 3 / 2 ρ dρ dφ , (4.115) leading to 35 35 Using ˆ a ρ = ˆ a x cos φ + ˆ a y sin φ so that its integral leads to zero. ¯ B (¯ r ) = μ 0 J 0 b 4 π 2 π 0 b a ˆ a ρ z + ˆ a z ρ [( ρ ) 2 + z 2 ] 3 / 2 dρ dφ (4.116) = ˆ a z μ 0 J 0 b 4 π 2 π 0 b a ρ [( ρ ) 2 + z 2 ] 3 / 2 dρ dφ (4.117) = ˆ a z μ 0 J 0 b 4 π 2 π − 1 ( ρ ) 2 + z 2 b a = ˆ a z μ 0 J 0 b 2 1 √ a 2 + z 2 − 1 √ b 2 + z 2 . (4.118) Furthermore, if the total current flowing through the ring is defined as I 0 = J 0 ( b − a ), we have 36 36 In this final form, it is possible to take a limit a → b as lim b → a { ¯ B (¯ r ) } = ˆ a z μ 0 I 0 2 a a 2 + z 2 lim b → a √ b 2 + z 2 − √ a 2 + z 2 b − a = ˆ a z μ 0 I 0 2 a a 2 + z 2 lim b → a b/ √ b 2 + z 2 1 = ˆ a z μ 0 I 0 2 a a 2 + z 2 a √ a 2 + z 2 = ˆ a z μ 0 I 0 2 a 2 [ a 2 + z 2 ] 3 / 2 , which is consistent with Eq. (2.260). ¯ B (¯ r ) = ˆ a z μ 0 I 0 2 b b − a 1 √ a 2 + z 2 − 1 √ b 2 + z 2 (4.119) = ˆ a z μ 0 I 0 2 b b − a √ b 2 + z 2 − √ a 2 + z 2 ( a 2 + z 2 )( b 2 + z 2 ) . (4.120) z J 0 8 8 a A -Figure 4.14 A steady volume electric current density flowing in a cylindrical region of radius a in the z direction. The magnetic vector potential is known inside the cylinder.
eBook - PDFGeophysical Field Theory and Method, Part A
Gravitational, Electric, and Magnetic Fields
- Alexander A. Kaufman(Author)
- 1992(Publication Date)
- Academic Press(Publisher)
Thus, measuring the magnetic field at the borehole axis we can in principle study the change of the casing parameter n l , which is defined by its thickness, radius, and magnetic permeability. Let us also notice that 550 IV Magnetic Fields due to the axial symmetry the molecular currents have an azimuthal component i
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