Download Biot-Savart Law

Magnetostatics
There are two major laws governing magnetostatics:
1. Biot-Savart law
2. Ampere’s circuit law
Just as Gauss’s law is special case of Coulombs law,
Ampere’s law is a special case of Biot-Savart’s law.
We know, Coulomb's law relates electric fields to the point
charges which are their sources. In a similar manner, BiotSavart Law relates magnetic fields to the currents which
are their sources.
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Derivation of Gauss’ Law using Coulombs law

Consider a sphere drawn around a positive point charge.
Evaluate the net flux through the closed surface.
Net Flux =


E  dA 

E cos dA 

EdA
kq
For a Point charge E  2
r
kq
   EdA   2 dA
r
kq
kq
  2  dA  2 (4 r 2 )
r
r
  4 kq
4 k 
1
0
 net 
where  0  8.85x10
qenc
0
E n
cos 0  1
nˆ
dA
12
Gauss’ Law
dA  nˆdA

C2
Nm 2

2
Biot-Savart Law
An electric current produces a magnetic field. Consider a small
segment of wire with length dl carrying a current I. This small
segment will produce a small magnetic field dB at a point P whose
magnitude and direction are given by the Biot-Savart law.
o I dl * r̂
dB 
4 r 2
r̂ is a unit vector
Specifying the direction of r ( distance from the
current to the field)
and µ0 is the permeability.
The Biot-Savart Law is used to calculate
the magnetic field at a given position.
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2. Gauss’ law for Magnetic Fields
Gauss’ law for magnetic fields is a formal way of saying that
magnetic monopoles do not exist.
The law states that the net magnetic flux B through any closed
surface must be zero.
   B dS  0
B
** Maxwell’s second equation
For comparison, Gauss’ law for electric fields
   E dS 
E
Qenc
0
In both equations, the integral is taken over a closed Gaussian surface.
E 

o
4
Ampere’s Circuital Law in Integral Form
• It states that “the circulation of the magnetic flux density
in free space over a closed path is proportional to the
total current enclosed in the surface.”
 B  dl   I
0 encl
C
Iencl is current through S:
I encl   J  dS
S
 B  dl    J  dS
0
C
S
Where J is defined as current density8
Ampere’s Circuital Law in Differential form
Applying Stroke's theorem to left-hand side of above equation
 B  dl     B  dS    J  dS
0
L
S
S
Comparing the surface integrals in above expressions
  B  0 J
Note: ** Ampere’s circuital
Law is Maxwell’s IV equation.
H  J  0
This shows that magnetostatic
field is nonconservative in nature.
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3. Faraday’s Law **
Changing magnetic field gives rise to electric current.
Induced emf if the loop is open-circuited
 B
 
t
Loop
E
EMF(Ve)
B (increasing)
Induced emf in the loop is  
>
 E  dl
P
^
I
<
Field by I
Integral form
Differential form
 B
P E  d l    t
B
 E  
t
Note: ** Faraday’s Law is Maxwell’s III equation.
B (increasing)
^
Four fundamental Laws:
1.
2.
 E  dS 
S
Qenclosed
o
 B  dS  0
S
3.
4.
 B
 E  dl   t
P

B  dl   0 I encl
Faradays
Amp Law
P
Asymmetry in the above laws
1. In equation 1 and 2, the R.H.S. of equation 2 contains no value,
because of the non existence of the magnetic monopole.
2. In equation 3 changing magnetic flux gives rise to electric field.
There is no such corresponding term relating to changing electric flux
producing magnetic field in equation 4. This led Maxwell to introduce a
new term corresponding to changing electric flux in equation 4.
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The corresponding term is
 0 0
 E  dl  
 E
t
 B  dl  0 I encl  0 0
P
P

 E
 0 ( I encl  I d )
t
I d  displacement curent   0
 B
t
B  dl   0 I encl
P
 E
t
Maxwell’s Equations in Integral Form
1.
Qenclosed
E

d
S


o
3.
P
S
2.
 B  dS  0
 E  dl  
4.
 B
t

E 
B

d
l


I



0
P
0  encl
t 

S
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CONTINUITY EQUATION
From the principle of charge conservation, the rate of decrease of charge
within a given volume must be equal to the net current flowing out
through the surface of the volume. Thus current Iout coming out of the
closed surface is
 dQin
I out   J  dS 
dt
where Qin is the total charge enclosed by the closed surface.
Using the divergence theorem
 J  dS     Jdv
S
and
v
dQin
d
v

   v dv   
dv
dt
dt v
v t
v
dv
   Jdv   
v
v t
v
J  
t
13
This equation is called continuity equation. It is derived from principle of
conservation of charge and shows that there can not be accumulation
of charge at any point.
For steady currents,
v
0
t
 J  0
Hence the total charge leaving a volume is the same as the total charge entering
it.
14
Maxwell’s modification of Ampere’s Law In
Differential Form
 B  dl   0 I encl
From Ampere’s Law
where
P
I encl   J  dS
S
Applying Stroke's theorem to left-hand side of above equation
   B  dS  0  J  dS
S
Differential form of Ampere’s Law
S
  B  0J
15
Taking divergence of above equation
    B   0   0   J 
Since R.H.S. of above equation is not zero**, so let us apply Continuity
equation and Gauss’s law (Electrostatics) we get
 v

 E 
J  
   0  E       0

t
t
 t 
 E 
  J    0
0
 t 
Hence
E 

or .  J   0
0
t 

  B   0 (J   0
E
)
t
E
E
The term  0
is called as displacement current density . J d   0
t
t
Hence modified Ampere’s Law
(differential form)
  B   0 J  J d 
**Note: Fundamental theorem of vector analysis;
    B  0
16
Maxwell’s Equations in Integral Form
1.
2.
 E  dS 
S
Qenclosed
o
 B  dS  0
S
3.
4.
“Gauss Law in Electrostatics”; Electric flux
coming out from the surface of the body is
equivalent to the charge enclosed by the body.
“Gauss Law in Magnetostatics”; Magnetic
flux coming out from the closed surface of the
body is zero as no magnetic monopole exist.
“Faradey’s Law”; Changing Magnetic flux
produce electric current (or field) in aclosed

loop. where Magnetic flux B  B.ds
B
 E  dl    t
L

E

 B  dl   0  i   0 t
L

S



“Modified Ampere’s Law” or “MaxwellAmpere’s Law”; Changing electric flux can
produce magnetic field in a discontinuous
circuit to hold Ampere’s circuital Law.
 
where Electric flux E   E.ds
S
17
Maxwell’s Equations in Integral Form
1.
Qenclosed
E

d
S


o
3.
P
S
2.
 E  dl  
 B  dS  0
4.
 B
t

 E
 B  dl   0  i   0 t

P



S
Maxwell’s Equations in differential form
1.   E 
2.

0
B  0
3.
4.
E  -
B
t
  B   0 (J   0
E
)
t
20
Case 1: Maxwell equations in free space*
(no free charges and no currents)
  0, J  0, q  0, i  0
E  0
B  0
B
E  t
E
  B  0 0
t
* Helpful to understand Electromagnetic waves in free space.
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