PPT No. 21
* Magnetic Dipoles
* Force and Couple on a Dipole
* Energy in Magnetic Fields
Models of
Magnetic Dipoles and Magnetic Dipole Moments
There are two models for the description of
magnetic dipoles and moments as follows
A) Current Loop Model
B) Two Monopoles or Electric Dipole Model
A) Current Loop Model
A Magnetic Dipole is a closed circulation of Electric current
A single loop of wire with some constant current flowing
through it, has a Magnetic field and behaves like a dipole.
In the multipole expansion of the magnetic vector potential A
of a current loop, there is no monopole contribution
(as Magnetic monopoles are not found to exist in nature) &
the dipole term is the most dominant one at large distance.
The fundamental magnetic element is magnetic dipole.
Its field falls off in proportion to 1/r3.
A) Current Loop Model
A Magnetic Dipole
The magnetic moment is a vector quantity associated with
the magnetic properties of magnetic dipoles.
The magnetic moment μ of the current loop is equal to
the amount of current I flowing through the loop
multiplied by the area A encompassed by the loop, and
its direction is given by the right hand rule for rotations.
The magnetic moment μ of the current loop gives
its characteristics in essence..
Magnetic Moment- Current Loop Model
Fig Magnetic moment of a current loop
A) Current Loop Model
A Magnetic Dipole
The electron and other fundamental particles, generate
Magnetic field similar to that due to a current loop of wire.
It is due to its orbital motion and
Intrinsic property called as spin.
It behaves like a very minute magnetic dipole and
has a magnetic dipole moment.
The relationships for a finite current loop are applicable
to the magnetic dipoles due to electrons.
The magnitude of elementary particle's
intrinsic magnetic moment is a fixed number.
B) Two Monopoles or Electric Dipole Model
According to the Gilbert model,
a permanent magnet, such as a bar magnet,
has magnetic poles of equal magnitude but
opposite polarity called as the "North" and "South" poles.
The magnetic force produced by a bar magnet having
p strength of each pole & d the distance separating poles
at a given point in space,
is proportional to the product
B) Two Monopoles or Electric Dipole Model
The magnetic force is proportional to μ
where μ describes the "magnetic moment" or
"dipole moment" of the magnet along a distance R.
its direction is given by the angle
between R and the axis of the bar magnet.
It can be thought of as a vector pointing
from the South to the North of a magnetic dipole,
B) Electric Dipole Model
Though this situation and related equations are
analogous to that of Electric dipole and its Dipole moment,
it is not founded on natural basis.
(Electric monopoles exist, however,
magnetic monopoles are not found in nature)
B) Electric Dipole Model
It is shown that
even a permanent magnet owes its magnetism
to the intrinsic magnetic dipole moment of the electron.
Therefore it can be said that
the origin of magnetic dipoles is only in the mechanism of
current loops or quantum-mechanical spin.
The SI unit of μ has two equivalent representations:
1 m2·A = 1 J/T.
Force on a Magnetic Dipole
The force due to a magnetic dipole moment m
can also be written as follows
For a current loop model
For Pair of monopoles or
electric dipole model
They can be converted into each other by using the relation
Force & Couple on a Magnetic Dipole
Consider a conducting loop carrying a current I,
having length L and breadth W (area A= LW) and
making angle θ with the direction of magnetic B-field.
The forces acting on both the ends are given by
magnetic moment μ
Force & Torque on a Magnetic Dipole
Magnetic Force and Torque on a current loop in B-Field
Couple on a Magnetic Dipole
Two equal and opposite forces acting at the ends
constitute a couple or torque given by
Couple on a Magnetic Dipole
From the geometry of a current loop it is evident that
this torque tends to line up
the magnetic moment with
the Magnetic field B,
which represents its lowest energy configuration
Example:
the torque on a current-carrying coil in a DC Motor
makes the coil to rotate.
Energy in Magnetic Fields
The phenomenon of magnetism gives rise to
magnetic potential energy due to which
a magnetic object has the potential
to move other similar objects.
The potential energy of a Magnet of
Magnetic moment μ in a Magnetic field B is defined as
the work of magnetic torque for
re-alignment of the magnetic dipole moment vector.
Energy in Magnetic Fields
Using the expression for the magnetic torque
on a current loop
the expression for magnetic potential energy U
can be developed as
(μ = magnetic dipole moment,
B = Magnetic Field)
Energy in Magnetic Fields
The energy U is expressed as a scalar product.
The magnetic potential energy is lowest when
the magnetic moment is aligned with the magnetic field and
highest when magnetic moment
is pointing in a direction opposite to the magnetic field
Energy in Magnetic Fields
The difference in energy
between aligned and anti-aligned positions is
The amount of rotational work W
to rotate the current loop from angle 00 to 1800
These relationships for a finite current loop
are applicable to the magnetic dipoles of
Electron orbits and electron spin also