Page 1
Magnetic forces:
Page 2
Magnetic forces:
Scalar Magnetic Potential and its limitations:
In studying electric field problems, we introduced the concept of electric potential that simplified
the computation of electric fields for certain types of problems. In the same manner let us relate
the magnetic field intensity to a scalar magnetic potential and write:
...................................(18)
From Ampere's law , we know that
......................................(19)
Therefore, ............................(20)
But using vector identity, we find that is valid only where . Thus
the scalar magnetic potential is defined only in the region where . Moreover, Vm in
general is not a single valued function of position.
This point can be illustrated as follows. Let us consider the cross section of a
coaxial line as shown in fig 7.
Page 3
Magnetic forces:
Scalar Magnetic Potential and its limitations:
In studying electric field problems, we introduced the concept of electric potential that simplified
the computation of electric fields for certain types of problems. In the same manner let us relate
the magnetic field intensity to a scalar magnetic potential and write:
...................................(18)
From Ampere's law , we know that
......................................(19)
Therefore, ............................(20)
But using vector identity, we find that is valid only where . Thus
the scalar magnetic potential is defined only in the region where . Moreover, Vm in
general is not a single valued function of position.
This point can be illustrated as follows. Let us consider the cross section of a
coaxial line as shown in fig 7.
In the region , and
Fig. 7: Cross Section of a Coaxial Line
If Vm is the magnetic potential then,
If we set Vm = 0 at then c=0 and
We observe that as we make a complete lap around the current carrying conductor , we reach
again but Vm this time becomes
We observe that value of Vm keeps changing as we complete additional laps to pass through the
same point. We introduced Vm analogous to electostatic potential V. But for static electric fields,
and , whereas for steady magnetic field wherever
but even if along the path of integration.
Page 4
Magnetic forces:
Scalar Magnetic Potential and its limitations:
In studying electric field problems, we introduced the concept of electric potential that simplified
the computation of electric fields for certain types of problems. In the same manner let us relate
the magnetic field intensity to a scalar magnetic potential and write:
...................................(18)
From Ampere's law , we know that
......................................(19)
Therefore, ............................(20)
But using vector identity, we find that is valid only where . Thus
the scalar magnetic potential is defined only in the region where . Moreover, Vm in
general is not a single valued function of position.
This point can be illustrated as follows. Let us consider the cross section of a
coaxial line as shown in fig 7.
In the region , and
Fig. 7: Cross Section of a Coaxial Line
If Vm is the magnetic potential then,
If we set Vm = 0 at then c=0 and
We observe that as we make a complete lap around the current carrying conductor , we reach
again but Vm this time becomes
We observe that value of Vm keeps changing as we complete additional laps to pass through the
same point. We introduced Vm analogous to electostatic potential V. But for static electric fields,
and , whereas for steady magnetic field wherever
but even if along the path of integration.
Vector magnetic potential due to simple configurations:
We now introduce the vector magnetic potential which can be used in regions where
current density may be zero or nonzero and the same can be easily extended to time varying
cases. The use of vector magnetic potential provides elegant ways of solving EM field problems.
Since and we have the vector identity that for any vector , , we
can write .
Here, the vector field is called the vector magnetic potential. Its SI unit is Wb/m.
Thus if can find of a given current distribution, can be found from through a curl
operation. We have introduced the vector function and related its curl to . A vector
function is defined fully in terms of its curl as well as divergence. The choice of is made as
follows.
...........................................(23)
By using vector identity, ...........................................(24)
.........................................(25)
Great deal of simplification can be achieved if we choose .
Putting , we get which is vector poisson equation.
In Cartesian coordinates, the above equation can be written in terms of the components as
......................................(26a)
......................................(26b)
......................................(26c)
The form of all the above equation is same as that of
..........................................(27)
for which the solution is
..................(28)
Page 5
Magnetic forces:
Scalar Magnetic Potential and its limitations:
In studying electric field problems, we introduced the concept of electric potential that simplified
the computation of electric fields for certain types of problems. In the same manner let us relate
the magnetic field intensity to a scalar magnetic potential and write:
...................................(18)
From Ampere's law , we know that
......................................(19)
Therefore, ............................(20)
But using vector identity, we find that is valid only where . Thus
the scalar magnetic potential is defined only in the region where . Moreover, Vm in
general is not a single valued function of position.
This point can be illustrated as follows. Let us consider the cross section of a
coaxial line as shown in fig 7.
In the region , and
Fig. 7: Cross Section of a Coaxial Line
If Vm is the magnetic potential then,
If we set Vm = 0 at then c=0 and
We observe that as we make a complete lap around the current carrying conductor , we reach
again but Vm this time becomes
We observe that value of Vm keeps changing as we complete additional laps to pass through the
same point. We introduced Vm analogous to electostatic potential V. But for static electric fields,
and , whereas for steady magnetic field wherever
but even if along the path of integration.
Vector magnetic potential due to simple configurations:
We now introduce the vector magnetic potential which can be used in regions where
current density may be zero or nonzero and the same can be easily extended to time varying
cases. The use of vector magnetic potential provides elegant ways of solving EM field problems.
Since and we have the vector identity that for any vector , , we
can write .
Here, the vector field is called the vector magnetic potential. Its SI unit is Wb/m.
Thus if can find of a given current distribution, can be found from through a curl
operation. We have introduced the vector function and related its curl to . A vector
function is defined fully in terms of its curl as well as divergence. The choice of is made as
follows.
...........................................(23)
By using vector identity, ...........................................(24)
.........................................(25)
Great deal of simplification can be achieved if we choose .
Putting , we get which is vector poisson equation.
In Cartesian coordinates, the above equation can be written in terms of the components as
......................................(26a)
......................................(26b)
......................................(26c)
The form of all the above equation is same as that of
..........................................(27)
for which the solution is
..................(28)
In case of time varying fields we shall see that , which is known as Lorentz
condition, V being the electric potential. Here we are dealing with static magnetic field,
so .
By comparison, we can write the solution for Ax as
...................................(30)
Computing similar solutions for other two components of the vector potential, the vector
potential can be written as
......................................(31)
This equation enables us to find the vector potential at a given point because of a volume current
density . Similarly for line or surface current density we can write
...................................................(32)
respectively. ..............................(33)
The magnetic flux through a given area S is given by
.............................................(34)
Substituting
.........................................(35)
Vector potential thus have the physical significance that its integral around any closed path is
equal to the magnetic flux passing through that path.
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