Force in Magnetic fields and Magnetic Potential | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

 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.