When charges move relative to each other they experience a new
force in addition to the electrostatic one. This new force is the magnetic force
and can be shown to be a consequence of special relativity.
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The choice of basic element is not as obvious as the choice of the
point charge in electrostatics. There are a number of choices including the
current element, the moving charge and the magnetic dipole. The current element
turns out to be the most useful choice.
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The
fundamental magnetic laws can not be expressed as simply as the corresponding
electrostatic ones. Magnetic forces are not generally central ones and the force
between two current elements depends not only on their size and separation but
on their relative orientations as well.
Because of these complications the concept of a field (the
magnetic or B-field) is introduced at
the outset. In this way the treatment of magnetic interactions is split into two
parts
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The
definition of the B-field produced by
a current element and the calculation of the B-field due to more complex circuits
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The
magnetic force that acts on a current element when placed in a B-field.
From this the total force acting on more complex circuits can be calculated and
the force acting on a charge moving in a B-field
can be derived.
The
production of B-fields by steady currents: the Biot-Savart law
We require an expression for the B-field
due to a current element. The geometry is shown in the figure. We require the
field dB at a point P a distance r
from a current element of length dl
(described by a vector dl). The
current element carries a current I and the vector (r) from the current element to P makes an angle q to dl.
Experiments show that dB
is proportional to r-2, the
length of the current element (dl) and
the current (I) which flows and also sinq. Hence the field due to the current
element can be written as
The resultant field is found to be normal to the plane containing
the vectors r and dl so that the
equation for the field can be written as
where
is a unit vector along r.
This expression is known as the ‘Biot-Savart law’.
The strength of the magnetic interaction is defined by m0
( mu nought) which is known as the permeability of free space. m0
has a value of 4px10-7 Hm-1
(Henrys/metre).
The unit of B is
the tesla (T)
By use of a suitable integration the Biot-Savart law can be
used to calculate the B-field
resulting from a circuit L which can be decomposed into an infinite number of
connected current elements
Example:
B-field due to an infinitely long,
straight current carrying wire
Require field at P a perpendicular distance r from the wire
Consider dB
produced by an element of length dy which
lies a distance x from P
dB=m0Idysinq/(4px2)
but
y=r/tanq=rcosq/sinq
and
hence
dy=-rdq/sin2q
also
x=r/sinq
Substituting in for dy
and x
for an infinitely long wire the limits of the integral are p and 0. Hence
Þ
The resultant B-field is of the form of concentric circles around the wire
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The above application of the Biot-Savart law showed that the B-field
at a distance r from an infinitely
long, straight wire is
Consider
a circular path L around the wire. The path has a
length
2pr so the previous equation
can be written as B2pr=m0I or field x path length = m0I. This result can be shown to be a special case of a general
result.
‘the line integral of B
around a closed loop L is equal to the
algebraic sum of the currents which flow through the area bounded by L
multiplied by m0.’
This result is known as Ampère’s circuital law.
The
differential form of Ampère’s circuital law
By applying Stoke’s theorem the left hand side of the circuital
law can be changed from a line integral around the path L to a surface integral over the surface S enclosed by L
the total current flowing through S (the summation on the left hand
side of the equation) can be written in terms of a surface integral of the
current density J
hence Ampère’s circuital law can be written in the form
this result must be true for any choice of surface S
or at any point in space. Hence we obtain the general result
Ñ´B=m0J
This is the differential form of Ampère’s circuital law. This
equation also corresponds to part of one of Maxwell’s equations.
Magnetic
force acting on a current element
Experimentally the magnetic force dF acting on a current element dl
carrying a current I and placed in a uniform field B
is found to be given by
dF=BIdlsinq
the direction of the magnetic force is normal to the plane
containing both B and dl.
In vector notation
dF=Idl´B
For a real circuit L the
total magnetic force is given by the appropriate integral of the forces acting
on the individual current elements
Magnetic
forces between current elements and circuits
The B-field produced by
one current element produces a magnetic force which acts on the other element
(and vice-versa).
If the two elements are I1dl1 and I2dl2
and are separated by a distance r12
(unit vector
) then the force on I2dl2 due to I1dl1
is given by
if I1dl1 and I2dl2
are elements of circuits L1
and L2 respectively then
the total magnetic force acting between the circuits is given by
Example:
two infinite straight wires carrying currents I1
and I2 are parallel and a distance r
apart. The B-field at wire 2 due to wire 1 is m0I1/(2pr) so that the force on length l
of wire 2 is
If the two currents flow in the same direction then the force
between the wires is an attractive one, for opposite directions it is a
repulsive one.
Force
on a charged particle moving in B-field
The current I flowing in a current element can be
written in terms of the individual charge carriers using the equation J=nqv, where J is the
current density, n is the number
density of the charge carriers, q is
their individual charge and v is their
velocity. If the current element has length dl and cross sectional area A
then
I=JA and N=Adl
where N is the total
number of charge carriers contained within the current element. Hence
So the total magnetic force acting upon the current element can be
written
dF=BIdlsinq=BNqvsinq
From
which it can be seen that the force on each individual charge is
F=Bqvsinq
The resultant force is normal to both the field and the velocity of
the charge. Hence in vector notation the magnetic force on a charge particle is
F=qv´B
If an electric field E
is also present then the total force is the sum of the electrostatic and
magnetic forces
F=q(E+v´B)
Magnetic
Dipoles
Consider the torque acting upon a plane rectangular circuit carrying a current I.
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Force on each side of circuit is F=IaB and hence torque
T = force x perpendicular distance
T=2IaB(b/2)sinq = IabBsinq
= IABsinq
where A is the area of
the circuit.
As in the case of the electric dipole we can use vector notation to
express this result more concisely
T=IA´B
where A is the vector
corresponding to the area of the circuit and the resultant torque T is given by a vector which points
along the axis of rotation.
The previous result was derived for a
rectangular coil, however it can be extended to apply to a plane coil of any
shape. In the figure the large coil is divided into a series of rectangles. Each
rectangle experiences a torque given by IA´B where A is the area of the rectangle.
However the rectangles can be made infinitesimally small so that their ends
approximate to the shape of the circuit. The total torque acting on the circuit
is given by the sum of the torques acting on the individual rectangles which, if
the field is constant over the area of the circuit, is given by IA´B where A
is now the total area of the circuit.
Hence we have a general result that any circuit carrying a current I experiences a torque T=IA´B when placed in a magnetic field B.
This result is very similar to that derived for an electric dipole
placed in a uniform E-field (see
Lecture 4). Hence it is useful to
write the product IA as m
where m is the magnetic dipole moment
of the circuit. The magnetic torque is given by T=m´B
(cf T=p´E for an electric dipole).
Results derived for an electric dipole can be applied to the
present case of a magnetic dipole e.g. potential energy of a magnetic dipole
placed in a uniform magnetic field B
is –m.B
In addition it can be shown that the form of the B-field
produced by a current flowing around a circuit is very similar to the E-field
produced by an electric dipole.
Conclusion:
The
B-field produced by the flow of
charge around a circuit and the magnetic force on a circuit placed in a B-field
can only be described in terms of magnetic dipoles (or possibly higher
order poles). There is no equivalent of the single isolated charge of
electrostatics. Magnetic monopoles do not exist.
Magnetic fields can also be produced by certain (magnetic)
materials. In this case the B-field
results from the motion of the electrons around the nuclei of the material.
However the form of the resultant B-field
and the effect of applying a B-field
on the material are also consistent with the non-existence of magnetic
monopoles.
Consequences
of the non-existence of magnetic monopoles
In electrostatics charges act as sources or sinks of the E-field.
Because there is no magnetic equivalent of single charges there are no sources
or sinks of the B-field. Hence the
lines of B are continuous and have no start or end.
Gauss’s
Law for B-fields
For E-fields Gauss’s
law states in integral form that
(A)
‘the flux of E over
any closed surface is equal to the algebraic sum of the charges enclosed by the
surface divided by e0.’
In differential form Gauss’s law for E-fields has the form
where r is the free charge density.
However for the B-field
case there is no equivalent of the isolated charge, only magnetic dipoles which
can be thought of as consisting of two equal but opposite magnetic monopoles.
Hence the summation on the right hand side of Equation (A) will consist of an
equal number of positive and negative terms resulting in a total summation of
zero.
Hence for B-fields,
Gauss’s law has the form
(B)
or
(C)
Equations (B) and (C) are the integral and differential forms of
another of Maxwell’s equations.
Their form arises because of the non-existence of magnetic
monopoles.
The zeros on the right hand sides of (B) and (C) makes these
equations much less useful for finding the form of B-fields in comparison to the corresponding equation for E-fields.The main uses of
(B) and (C) is in summarising in a condensed form an important property of
magnetic fields.
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Conclusions
·
The
origin and treatment of magnetic (B)
fields
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The Biot-Savart law for finding B-fields due to currents
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Ampère’s Circuital Law: integral and differential form
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Magnetic force on a current carrying element
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Magnetic forces between current elements and circuits
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The magnetic force on moving charges
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The magnetic torque acting on a circuit
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The non-existence on magnetic monopoles and consequences
· Gauss’s law for B-fields: integral and differential form