Physics
Torque on Magnetic Dipole
The torque on a magnetic dipole is the rotational force experienced by a magnetic dipole when placed in an external magnetic field. It is given by the cross product of the magnetic moment vector and the magnetic field vector. The torque causes the magnetic dipole to align itself with the direction of the magnetic field.
Written by Perlego with AI-assistance
Related key terms
1 of 5
5 Key excerpts on "Torque on Magnetic Dipole"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
● If an external agent rotates a magnetic dipole from an initial orientation θ i to some other orientation θ f and the dipole is stationary both initially and finally, the work W a done on the dipole by the agent is W a = ΔU = U f – U i . The Magnetic Dipole Moment As we have just discussed, a torque acts to rotate a current-carrying coil placed in a magnetic field. In that sense, the coil behaves like a bar magnet placed in the magnetic field. Thus, like a bar magnet, a current-carrying coil is said to be a mag- netic dipole. Moreover, to account for the torque on the coil due to the magnetic field, we assign a magnetic dipole moment μ → to the coil. The direction of μ → is that of the normal vector n → to the plane of the coil and thus is given by the same right- hand rule shown in Fig. 28.7.2. That is, grasp the coil with the fingers of your right hand in the direction of current i; the outstretched thumb of that hand gives the direction of μ → . The magnitude of μ → is given by μ = NiA (magnetic moment), (28.8.1) 875 28.8 THE MAGNETIC DIPOLE MOMENT in which N is the number of turns in the coil, i is the current through the coil, and A is the area enclosed by each turn of the coil. From this equation, with i in amperes and A in square meters, we see that the unit of μ → is the ampere-square meter (A · m 2 ). Torque. Using μ → , we can rewrite Eq. 28.7.3 for the torque on the coil due to a magnetic field as τ = μB sin θ, (28.8.2) in which θ is the angle between the vectors μ → and B → . We can generalize this to the vector relation τ → = μ → × B → , (28.8.3) which reminds us very much of the corresponding equation for the torque exerted by an electric field on an electric dipole—namely, Eq. 22.7.3: τ → = p → × E → . In each case the torque due to the field—either magnetic or electric—is equal to the vector product of the corresponding dipole moment and the field vector.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
The magnitude of μ → is given by μ = NiA (magnetic moment), (28.8.1) in which N is the number of turns in the coil, i is the current through the coil, and A is the area enclosed by each turn of the coil. From this equation, with i in amperes and A in square meters, we see that the unit of μ → is the ampere-square meter (A · m 2 ). Torque. Using μ → , we can rewrite Eq. 28.7.3 for the torque on the coil due to a magnetic field as τ = μB sin θ, (28.8.2) in which θ is the angle between the vectors μ → and B → . We can generalize this to the vector relation τ → = μ → × B → , (28.8.3) which reminds us very much of the corresponding equation for the torque exerted by an electric field on an electric dipole—namely, Eq. 22.7.3: τ → = p → × E → . In each case the torque due to the field—either magnetic or electric—is equal to the vector product of the corresponding dipole moment and the field vector. Energy. A magnetic dipole in an external magnetic field has an energy that depends on the dipole’s orientation in the field. For electric dipoles we have shown (Eq. 22.7.7) that U(θ) = − p → ⋅ E → . In strict analogy, we can write for the magnetic case U (θ) = − μ → ⋅ B → . (28.8.4) In each case the energy due to the field is equal to the negative of the scalar prod- uct of the corresponding dipole moment and the field vector. A magnetic dipole has its lowest energy (= –μB cos 0 = –μB) when its dipole moment μ → is lined up with the magnetic field (Fig. 28.8.1). It has its highest energy (= −μB cos 180° = +μB) when μ → is directed opposite the field. From Eq. 28.8.4, with U in joules and B → in teslas, we see that the unit of μ → can be the joule per tesla (J/T) instead of the ampere-square meter as suggested by Eq. 28.8.1. Work. If an applied torque (due to “an external agent”) rotates a magnetic dipole from an initial orientation θ i to another orientation θ f , then work W a is done on the dipole by the applied torque.
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2023(Publication Date)
- Società Editrice Esculapio(Publisher)
If a steady direct current i flows in a coil when it is immersed in a uniform magnetic field B, it undergoes the action of an overall torque with respect to its centre of mass τ G = m × B = − NiA u n × B . Hence, when a current flows in the coil, it rotates. If the coil is con- nected to a spring which opposes the rotation by generating a torque ′ τ G = −c θ u z where c is the torsion constant of the spring, the needle stops in its equilibrium position where NiAB = c θ . The current intensity is therefore given by i = c NAB θ being proportional to the angular deviation of the needle. If the current flows in the oppo- site direction, the needle rotates in the opposite direction. N S 9.6 Magnetic Dipole Moment The torque undergone by a one-turn coil immersed in a uniform magnetic field is τ G = −iAB sin θ u z . If a vector m = iA u n is defined, it can be remarked that the torque is given by τ G = m × B = iA u n × B . The obtained torque expression is analogous to the one that holds for the torque acting on an electric dipole immersed in an electrostatic field τ G = p × E , with the replacements of p by m and E by B, so the quantity m is called magnetic di- pole moment. This analogy allows to assume a coil in which a current flows as a magnetic dipole, i.e. as the fundamental element of the magnetic interaction. This assumption is formally established by stating that: any flat one-turn coil of area A in which a steady current i flows, can be assumed to be an elementary magnetic dipole of magnetic dipole moment m = iA u n , where the direction of the normal to the coil surface must be chosen by applying the right- hand screw rule. 9.7 Work Done by a Magnetic Force The work done by a magnetic force for a displacement of a charge q that moves with velocity v in a static magnetic field B from point A to point B in the space is given by W = q v × B ( ) id r A B ∫ .
eBook - PDFOptical Tweezers
Methods and Applications
- Miles J. Padgett, Justin Molloy, David McGloin, Miles J. Padgett, Justin Molloy, David McGloin(Authors)
- 2010(Publication Date)
- Chapman and Hall/CRC(Publisher)
We can use the preceding discussion to elucidate the form of the electromagnetic torque acting on a dielectric medium. We start by considering the single dipole introduced in section 2.1. The force density f d gives the nett force on the centre of mass of our point dipole. The associated torque density, about the origin of coordinates, acting on this centre of mass is r × f d and to this we need to add the internal torque P × E , which acts to orient the dipole: t d = r × f d + P × E = r × (( d · ∇ ) E + ˙ d × B )δ( r − R ) + d × E δ( r − R ). (57) Integrating this over a volume containing the dipole gives the total torque T d = R × (( d · ∇ ) E + ˙ d × B ) + d × E . (58) We might expect that the force, f c , being based on the microscopic distribution of charges should also give the correct torque. In this case, however, there is no separation of a centre of mass and so the full torque density is t c = r × [ − E ( d · ∇ δ( r − R )) + ˙ d × B δ( r − R ) ] . (59) 260 Optical Tweezers: Methods and Applications (Paper 4.4) The total torque is again obtained by integrating over a volume containing the dipole: T c = − � r × E ( d · ∇ δ( r − R )) d V + � r × ( ˙ d × B )δ( r − R ) d V = � δ( r − R ) d · ∇ ( r × E ) d V + R × ( ˙ d × B ) = T d . (60) At the microscopic level, both force densities give the correct torque. The only distinction is that using f d requires us to add the contribution associated with orientation of the dipole. At the other extreme we would like to calculate the nett torque on a dielectric medium by integrating the torque density (57) or (59) expressed in terms of the macroscopic fields: t d = r × (( ¯ P · ∇ ) ¯ E + ˙ ¯ P × ¯ B ) + ¯ P × ¯ E t c = r × (( −∇ · ¯ P ) ¯ E + ˙ ¯ P × ¯ B ).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
Any non-uniform magnetic field whether caused by permanent magnets or by electric currents will exert a force on a small magnet in this way. Mathematically, the force on a small magnet having a magnetic moment m due to a magnetic field B is: ______________________________ WORLD TECHNOLOGIES ______________________________ where the gradient ∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B . To understand this equation, note that the dot product m · B = mB cos( θ ), where m and B represent the magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B ). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions having their own m then summing up the forces on each of these regions. Torque on a magnet due to a B-field Magnetic torque on a magnet due to an external magnetic field can be observed by placing two magnets near each other while allowing one to rotate. Magnetic torque is used to drive simple electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of electromagnets. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft. Magnetic torque τ tends to align a magnet's poles with the B-field lines (since m is in the direction of the poles this is equivalent to saying that it tends to align m in the same direction as B ). This is why the magnetic needle of a compass points toward earth's north pole.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
