Magnetic Dipole Theory and Magnetic Potentials
To understand magnetic dipole moment, you first need to understand the simple magnetic moment.
Magnetic movement:
It is the magnitude that signifies the magnetic orientation and strength of a magnet or other object that creates a magnetic field. Examples of such objects that have magnetic moments contain loops of electric current (like electromagnets), elementary particles (such as electrons), permanent magnets, various molecules, and several astronomical objects (such as various planets, certain moons, stars, etc).
Exactly, the word magnetic moment normally denotes to a system's magnetic dipole moment, the component of the magnetic moment that can be denoted by the same magnetic dipole: magnetic north and south pole divided by a very minor distance. The magnetic dipole component is enough for small adequate magnets or for large sufficient distances. Higher-order expressions (such as the magnetic quadrupole moment) can be needed in addition to the dipole moment for prolonged objects.
The magnetic dipole moment of an object or thing is readily defined in relation to the torque that the object experiences in a certain magnetic field. The same applied magnetic field generates larger torques on objects with bigger magnetic moments. The direction and strength of this torque depend not only on the degree of the magnetic moment but also on its location relative to the direction of the magnetic field. The magnetic moment can also be considered, so, to be a vector.
Definition
It can be defined as a vector linking the aligning torque on the object from an outside applied magnetic field to the field vector itself. The relationship is written by
\[tau = m\times B\]
Where τ is the torque acting on the dipole, B is the outside magnetic field, and m is on the magnetic moment.
This definition is based on the principle, of measuring the magnetic moment of an unknown sample. For a current loop, this definition hints at the magnitude of the magnetic dipole moment corresponding to the product of the current times the region of the loop. More, this definition agrees to the calculation of the projected magnetic moment for any known macroscopic current sharing.
A substitute definition is helpful for thermodynamic calculations of the magnetic moment. In this definition, the magnetic dipole moment of a structure is the negative gradient of its fundamental energy, Uint, with respect to the outer magnetic field:
\[m = \frac{-\widehat{x} \partial U{int}}{\partial B{x}} - \frac{\widehat{y} \partial U{int}}{\partial B{y}} - \frac{\widehat{z} \partial U{int}}{\partial B{z}}\].
Magnetic Dipole
It is the magnitude that signifies the magnetic orientation and strength of a magnet or other object that yields a magnetic field. Exactly, a magnetic moment mentions to a magnetic dipole moment, the constituent of the magnetic moment that can be signified by a magnetic dipole. A magnetic dipole is a magnetic north pole and South Pole divided by a minor distance.
Magnetic dipole moments have sizes of the current time’s region or energy separated by magnetic flux density. The unit for dipole moment in centimeter–gram–second electromagnetic system, in meter–kilogram– second–ampere is an ampere-square meter, is the erg (unit of energy) per gauss (unit of magnetic flux density). One thousand ergs per gauss equal to one ampere-square meter.
The theory underlying the magnetic dipole
The field (magnetic) of any magnet can be exhibited by a series of terms for which every term is more complicated (having finer angular features) than the one before it. The first three terms of that series are known as monopole (denoted by isolated magnetic south or north pole) the dipole (denoted by two equal and opposite magnetic poles), and the quadrupole (denoted by four poles that combine together form two equal and opposite dipoles). The degree of the magnetic field for every term reduces progressively sooner with distance than the previous term so that at big enough distances the first non-zero term will govern.
For several magnets, the first non-zero word is the magnetic dipole moment. (To date, no isolated magnetic monopoles have been experimentally identified.) A magnetic dipole is the boundary of either a current loop or a pair of poles as the dimensions of the source are drop to zero while keeping the moment continuous. As long as these restrictions only apply to fields far from the sources, they are the same. However, the two models give different predictions for the inside field
Magnetic potentials
Usually, the equations for the magnetic dipole moment (and higher-order terms) are derived from theoretical quantities known as magnetic potentials which are simpler to deal with mathematically than the magnetic fields.
In the magnetic pole model, the related magnetic field is the demagnetizing field H{\displaystyle \mathbf {H} }. Then the demagnetizing portion of H does not include, by description, the part of H {\displaystyle \mathbf {H} }due to free currents, there occurs a magnetic scalar potential such that
\[H(r) = - \triangledown \psi\]
In the amperian loop model, the related magnetic field is the magnetic induction B{\displaystyle \mathbf {B} }. Since magnetic monopoles do not occur, there happens a magnetic vector potential such that
\[B(r) = \triangledown\times A.\]
Both of these potentials can be measured for any arbitrary current sharing (for the amperian loop model) or magnetic charge distribution (for the magnetic charge model) is providing that these are restricted to a small adequate region to give:
\[A(r, t) = \frac{\mu_{0}}{4\pi} \int {j(r')}{|r - r'|} dV'\]
\[\psi (r, t) = \frac{1}{4\pi} \int {\rho(r')}{|r - r'|} dV'\]
Here p is the magnetic pole strength density in analogy to the electric charge density J is the current density in the amperian loop model, which leads to the electric potential, and the volume (triple) integrals over the coordinates that make up r’. The denominators of this equation can be prolonged with the help of multipole expansion to give a sequence of terms that have greater power of distances in the denominator. The first nonzero term, so, will dominate for great distances. The first non-zero term for the vector potential is given by
\[A(r) = \frac{\mu_{0}}{4\pi} \frac{m\times r}{|r|^{3}}\]
Where m is :
\[m = \frac{1}{2} \iiint{V} r\times j dV\]
Here r is the position vector, j is the electric current density & the integral is a volume integral.
× is the vector cross product, In the magnetic pole perspective, and the first non-zero term of the scalar potential is
\[\psi (r) = \frac{m.r}{4\pi |r|^{3}}\].
Here m may be represented in terms of the magnetic pole strength density but is more usefully expressed in terms of the magnetization field as:
\[m = \iiint M dVm\].
The same symbol m is used for both equations since they produce equivalent results outside of the magnet.
Derivation of Magnetic Dipole Moment Formula
Magnetic Dipole moment-
The magnetic field, R at a distance l along its axis, B due to a current loop carrying current i of radius, is given by:
\[B = \frac{\mu_{0}i R^{2}}{2(R^{2} + l^{2})^{\frac{3}{2}}}\]
At the present, if we think a point which is far from the current loop such that l>>R, then we can estimate the field as:
\[B = \frac{\mu_{0}i R^{2}}{2l^{3}((\frac{R}{i})^{2} + 1)^{\frac{3}{2}}} \approx \frac{\mu_{0}i R^{2}}{2l^{3}} = \frac{\mu_{0}}{4\pi} \frac{2i(\pi R^{2})}{l^{3}}\]
currently, the area of the loop, A is
\[A = \pi R^{2}\]
therefore, the magnetic field can be written as
\[B = \frac{\mu_{0}}{4\pi} \frac{2iA}{l^{3}} = \frac{\mu_{0}}{4\pi} \frac{2\mu}{l^{3}}\]
We can mark this new quantity μ as a vector that points next to the magnetic field, so that
\[\bar{B} = \frac{\mu_{0}}{4\pi} \frac{2 \bar{\mu}}{l^{3}}\]
take in the astounding connection to the electric dipole field:
\[\bar{E} = \frac{1}{4 \pi \epsilon_{0}} \frac{2 \bar{p}}{r^{3}}\]
Thus we call this quantity μ→ the magnetic dipole moment. Different electric fields such as magnetic fields do not have ‘charge ‘counterparts. Hence there are no sources or sinks of magnetic fields, here can only be a dipole. Whatever we produce a magnetic field comes with both a sink and a source that is there is both a north pole and south pole. With the help of different ways, the magnetic dipole is the fundamental unit that can produce a magnetic field.
Most elementary particles behave fundamentally as magnetic dipoles. For instance, the electron itself behaves as a magnetic dipole and has a Spin Magnetic Dipole moment. This magnetic moment is inherent from the electron has neither an area A (it is a point object) nor does it spin around itself but is fundamental to the nature of the electron’s existence.
We can simplify the magnetic moment for ‘N’ turns of the wire loop as
μ = NiA
The magnetic field lines of a current loop look alike to that of an idealized electric dipole:
The internal magnetic field of a dipole
The two types of dipole (current loop and magnetic poles) give the same prediction for the magnetic field far from the source. On the other hand, inside the source region, they give different predictions. The magnetic field between both poles (see figure for Magnetic pole definition) is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while within a current loop it is in the same direction (see the figure to the right). The restrictions of these fields must also be diverse as the sources minimize to zero sizes. This difference only matters if the dipole limit is used to analyze fields inside a magnetic material.
If a magnetic dipole is shaped by making a present loop smaller and smaller, but keeping the product of current and area even, the limiting field is
\[B(x) = \frac{\mu_{0}}{4\pi} [\frac{3n(n.m) - m}{|x|^{3}} + \frac{8\pi}{3} m \partial (x)]\]
Contrasting the expressions in the previous section, this limit is exact for the internal field of the dipole.
If a magnetic dipole is formed by taking a "north pole" and a "south pole", bring them closer and closer together but keeping the product of magnetic pole charge and distance constant, the limiting field is
\[H(x) = \frac{1}{4\pi} [\frac{3n(n.m) - m}{|x|^{3}} + \frac{4\pi}{3} m \partial (x)]\]
These fields are connected by B = μ0(H + M), where M(x) = mδ(x) is the magnetization.
FAQs on Magnetic Dipole Moment
1. What is meant by magnetic dipole moment, and what is its SI unit?
The magnetic dipole moment is a vector quantity that represents the strength and orientation of a magnet or any object that produces a magnetic field. It is a measure of the object's tendency to align with an external magnetic field. The SI unit for magnetic dipole moment is Ampere-meter squared (A·m²). It is often denoted by the symbol M or µ.
2. What is the formula for the magnetic dipole moment of a current loop?
For a planar loop of wire carrying an electric current, the formula for the magnetic dipole moment (M) is given by:
M = N × I × A
Where:
- N is the number of turns in the loop.
- I is the current flowing through the loop in Amperes.
- A is the area of the loop in square meters.
3. How does a magnetic dipole fundamentally differ from an electric dipole?
The primary difference between magnetic and electric dipoles lies in their source and field structure:
- Source: An electric dipole consists of two equal and opposite stationary electric charges (monopoles) separated by a small distance. A magnetic dipole, however, is typically generated by a moving charge, such as a current in a loop, or by the intrinsic spin of elementary particles.
- Magnetic Monopoles: Isolated magnetic poles (monopoles) have never been observed in nature. This means you cannot have a separate 'north' or 'south' pole. Electric monopoles (positive and negative charges) exist freely.
- Field Lines: Electric field lines originate from positive charges and terminate on negative charges. In contrast, magnetic field lines always form continuous, closed loops, never starting or ending at a point.
4. How does a revolving electron in an atom create a magnetic dipole moment?
An electron revolving around the nucleus in an atom behaves like a tiny current loop. Since the electron carries a negative charge, its motion constitutes an electric current in the direction opposite to its revolution. According to Ampere's law, any current loop generates a magnetic field. This gives rise to an orbital magnetic dipole moment. The magnitude of this moment is directly proportional to the electron's angular momentum, making the atom behave like a tiny magnet.
5. What is the physical significance of the torque experienced by a magnetic dipole in a uniform magnetic field?
The torque experienced by a magnetic dipole in a uniform magnetic field is physically significant as it is the principle that governs the operation of many devices. The torque, given by the formula τ = M × B, is a rotational force that attempts to align the dipole's magnetic moment (M) with the external magnetic field (B). This principle is the basis for:
- Electric Motors: Where torque on a current-carrying coil causes it to rotate.
- Compasses: The needle, a magnetic dipole, aligns with the Earth's magnetic field.
- Galvanometers: Used to detect electric currents by measuring the torque and resulting deflection of a coil.
6. Why is a bar magnet often treated as an equivalent solenoid for calculations?
A bar magnet is considered an equivalent solenoid because their magnetic field patterns are remarkably similar, especially at points far from them. The reasoning stems from the atomic level: in a bar magnet, the magnetic moments of countless atoms align, creating microscopic current loops. The net effect of these atomic currents is a surface current that flows around the magnet, similar to the current in a solenoid. This equivalence is a powerful concept in physics because the magnetic field of a solenoid is easier to calculate mathematically, allowing physicists to model the behaviour of a bar magnet with greater precision.
7. Since isolated magnetic monopoles do not exist, what does this imply about the fundamental nature of magnetism?
The absence of magnetic monopoles is a fundamental principle of electromagnetism, stated by Gauss's law for magnetism (∇·B = 0). This implies that magnetic fields are fundamentally dipolar. Unlike electric fields that can start or end on a charge, magnetic field lines are always continuous and form closed loops. This tells us that magnetism is not caused by static 'magnetic charges', but is an intrinsic property related to the motion and spin of electric charges. Every source of a magnetic field, from a simple bar magnet to an elementary particle like an electron, must have both a north and a south pole.
