This document contains lecture notes on electromagnetic theory from a course taught by Arpan Deyasi. It discusses the Biot-Savart law, which gives mathematical expressions for the magnetic field created by steady current-carrying wires and distributions of electric current. It also covers the Lorentz force law and how it relates to the combined electric and magnetic forces on a moving charged particle. Examples are presented on calculating magnetic fields and forces. The document concludes by deriving the solenoidal property of magnetic fields.

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 – Biot-Savart Law
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2. Biot-Savart Law
In either presence or absence of magnetic material, magnetic field for a steady line
current flowing around a closed wire is given by
( )
0
0
3
0
dl r r
B i
4 r r
 −

=
 −

O
P i
r
0
r
where ‘i’ is line current density
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0
r r
−
Arpan Deyasi
Electromagnetic
Theory

3. Biot-Savart Law
( )
0
0
3
0
dS r r
B K
4 r r
 −

=
 −

( )
0
0
3
0
J r r
B dv
4 r r
 −

=
 −

For surface current
where ‘K’ is line current density
For volume current
where ‘J’ is line current density
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4. Lorentz Force
A point charge at rest, produces electrostatic field in its surroundings,
corresponding force is given by
elec
F qE
=
If the charge is allowed to move with uniform velocity, then corresponding
magnetostatic force
( )
mag
F q v B
= 
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5. Composite force
net elec mag
F F F
= +
( )
net
F qE q v B
= + 
( )
net
F q E v B
 
= + 
 
Lorentz Force
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6. Problem 1
Show that magnetic force does not work
Soln
We consider a point charge ‘q’ is moving with uniform velocity ‘v’, and traverses a
distance ‘dl’ In time ‘dt’ under the electromagnetic field
dl vdt
=
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7. ( )
dW q v B .vdt
= 
dW 0
=
Work done by magnetic force
mag
dW F .dl
=
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8. A charged particle moves with a uniform velocity 4i m/sec in a region where
V/m and Wb/m2. Find magnetic field so that velocity of the particle remains
constant.
( )
net
F q E v B
 
= + 
 
According to the Lorentz force
Problem 2
Soln
ˆ
E 20j
=
0
ˆ
B B k
=
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9. 0
ˆ ˆ ˆ
20j 4B i k 0
+  =
0
B 5
=
As per the condition
net
F 0
=
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10. Solenoidal property of magnetic field
From Biot-Savart law, magnetic field for a linear line current is
( )
0
0
3
0
J r r
B dv
4 r r
 −

=
 −

where magnetic field B is a function of sink coordinate i.e.,
( )
B B x, y,z
=
Current density J is a function of source coordinate i.e.,
( )
J J x ', y',z'
=
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11. Solenoidal property of magnetic field
Applying divergence
( )
0
0
3
0
J r r
.B . dv
4 r r
 
 −

 =   

 
−
 

0
3
J r
.B . dv
4 r
 
 
 =   
  

In simplified form
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12. 0
3
J r
.B . dv
4 r
 
 
 = 
 
  

Now
( )
3 3 3
J r r r
. J J.
r r r
 
  
 =  − 
   
   
Solenoidal property of magnetic field
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13. Solenoidal property of magnetic field
Since J does not depend on source coordinates
J 0
 =
Again
3
r
0
r
 =
3
J r
. 0
r
 

  =
 
 
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14. Solenoidal property of magnetic field
.B 0
 =
Significance: [i] Magnetic field through any closed
surface is zero
[ii] Lines of B does not have sources and are continuous
[iii] There are no magnetic charges which would
generate magnetic field
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