Magnetic Scalar and Vector Potentials

1. Course: Electromagnetic Theory
paper code: EI 503
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
Topic: Magnetostatics – Magnetic Potential
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2. 03-12-2021 Arpan Deyasi, EM Theory 2
Magnetic Scalar Potential
Magnetic field is related with scalar potential by the relation
m
B = −
where φm is the magnetic scalar potential
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3. 03-12-2021 Arpan Deyasi, EM Theory 3
Property of Magnetic Scalar Potential
We know from solenoidal property of magnetic field that
.B 0
 =
( )
m
. 0
 − =
2
m 0
  =
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4. 03-12-2021 Arpan Deyasi, EM Theory 4
Problem 1
Calculate magnetic scalar potential for infinite solenoid
Soln
ˆ
B NIz
= 
For infinite solenoid
Where ‘N’ is the umber of turns, ‘I’ is current in the winding
m
B = −
Also magnetic scalar potential is related with field as
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5. 03-12-2021 Arpan Deyasi, EM Theory 5
m m m
1
ˆ ˆ
ˆ ˆ
NIz z
z
 
  
 = −  + +
 
   
 
m
NI
z

= −

m NIz
 = −
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6. Magnetic Vector Potential
We know from solenoidal property of magnetic field that
.B 0
 =
Let A be a differentiable vector
( )
. A 0
  =
Comparing
B A
= 
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7. Expression of Magnetic Vector Potential
According to Biot-Savart law
0
3
J r
B dv
4 r
 
=
 
Now
3
1 r
r r
 
 = −
 
 
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8. Expression of Magnetic Vector Potential
0 1
B J dv
4 r
  
 
 =  
 
  
 

Now
( )
J 1 1
J J
r r r
   
 =  +  
   
   
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9. Expression of Magnetic Vector Potential
Since
J 0
 =
J 1
J
r r
   
  =  
   
   
J 1
J
r r
   
 = − 
   
   
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10. Expression of Magnetic Vector Potential
0 J
B dv
4 r
 
 

 =  
 
  
 

0 J
B dv
4 r
 

=   
  

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11. Comparing with
B A
= 
0 J
A dv
4 r
 

=  
  

We get
Expression of Magnetic Vector Potential
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12. Property of magnetic vector potential: Solenoidal
0 J
A dv
4 r
 

=  
  

0 J
.A . dv
4 r
 

 =   
  

0 J
.A . dv
4 r
 

 =   
  

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13. Property of magnetic vector potential: Solenoidal
Applying divergence theorem
0 J
.A dS
4 r

 =
 
By expanding the surface, we can make
.A 0
 =
So, magnetic vector potential is solenoidal
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14. Problem 2
Calculate M.V.P in the region surrounding an infinitely long, straight, filamentary current I
Soln
For long straight wire carrying current I
enc
I ˆ
B
2


= 

enc
I ˆ
A
2

 = 

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15. enc
z
A I
A
z 2

 

− =
  
Since the filament is uniform along Z direction, so A should not be a function of Z
enc
z I
dA
d 2

 − =
 
Let at
0 z
A 0
 =  =
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16. enc 0
z
I
A ln
2
 
 
=  
 
 
enc 0
z
I
ˆ
A ln z
2
 
 
=  
 
 
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17. B J
 = 
Magnetic vector potential is given by
B A
= 
( )
B A
 =  
Ampere’s circuital law is given by
Property of magnetic vector potential: Laplacian
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18. Property of magnetic vector potential: Laplacian
Comparing
( )
A J
  = 
( ) 2
.A A J
  − = 
Since
.A 0
 =
2
A J
  = −
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19. Problem 3
Magnetic vector potential is given by
2
ˆ
A z Wb / m
4

= −
Calculate total magnetic flux crossing the surface 1m<ρ<2m, φ=π/2, 0<z<5m
Soln
z
A ˆ
B A

=  = − 

2
ˆ ˆ
B
4 2
 
  
= − −  = 
 
  
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20. Magnetic flux
m B.dS
 = 
5 2
m
z 0 1
ˆ ˆ
.d dz
2
= =

 =   
 
5 2
m
z 0 1
dz d
2
= =

 = 
 
m 3.75 Wb
 =
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