Biot-Savart Law and Ampere's Circuital Law

Contributed by:

Jonathan James Thu, Jan 20, 2022 09:36 PM UTC

Biot-Savart Law, Magnetic dipole moment, Potential of a dipole sheet, Ampere's Circuital Law, Solenoid, Relative permeability

1. 4). Ampere’s Law and Applications
• As far as possible, by analogy with Electrostatics
• B is “magnetic flux density” or “magnetic induction”
• Units: weber per square metre (Wbm-2) or tesla (T)
• Magnetostatics in vacuum, then magnetic media
based on “magnetic dipole moment”

2. Biot-Savart Law 
• The analogue of Coulomb’s Law is dB(r)
the Biot-Savart Law r-r’
r
• Consider a current loop (I)
O r’
dℓ’
• For element dℓ there is an
o  d' x(r  r' )
associated element field dB dB(r )  3
4 r  r'
dB perpendicular to both dℓ’ and r-r’
same 1/(4r2) dependence
o is “permeability of free space”
defined as 4 x 10-7 Wb A -1 m-1 o  d' x(r  r' )
B(r ) 
4  r  r' 3
Integrate to get B-S Law 

3. B-S Law examples
I
(1) Infinitely long straight conductor dℓ 
dℓ and r, r’ in the page r’ z r - r’
dB is out of the page
O  dB
B forms circles r
centred on the conductor  r
sin  = cos  2 2 1/2
Apply B-S Law to get: r  z 
o I
B
2 r
B

4. B-S Law examples
(2) “on-axis” field of circular loop
dℓ
Loop perpendicular to page, radius a
r - r’ dB
dℓ out of page and r, r’ in the page r’ 
I r
On-axis element dB is in the page, a z dBz
perpendicular to r - r’, at  to axis.
Magnitude of element dB
 o I d  o I d a a
dB  2
 dB z  2
cos cos   1/2
4 r - r ' 4 r - r ' r - r' 
a2  z2 
Integrating around loop, only z-components of dB survive
The on-axis field is “axial”

5. On-axis field of circular loop
dℓ
o I r - r’ dB
Bon axis dB z  2
cos d
4 r - r ' r’ 
I r
o I o Ia 2 a z dBz
 2
cos  2 a   3
4 r - r ' 2 r - r'
Introduce axial distance z,
o I a 2
where |r-r’|2 = a2 + z2 Bon axis  3
2a  z
2 2
 2
2 limiting cases:
2
z 0 o I z a  o Ia
Bon  axis  and Bon  axis 
2a 2z 3

6. Magnetic dipole moment
The off-axis field of circular loop is
much more complex. For z >> a it is
identical to that of the electric dipole
p
E
4 or 3
2 
cos  ˆ
r  sin ˆ 
 om
 B
4 r 3

2 cos rˆ  sin ˆ
where m  a 2 I  I or m  a 2 I zˆ
 area enclosed by current loop
m
m “current times area” vs p “charge times distance”  r

7. B field of large current loop
• Electrostatics – began with sheet of electric monopoles
• Magnetostatics – begin sheet of magnetic dipoles
• Sheet of magnetic dipoles equivalent to current loop
• Magnetic moment for one dipole m = I  area 
for loop M = I A area A
• Magnetic dipoles one current loop
• Evaluate B field along axis passing through loop

8. B field of large current loop
• Consider line integral B.dℓ from loop
• Contour C is closed by large semi-circle which contributes
zero to line integral
I (enclosed by C)
a
z→-∞ C z→+∞
o I  a 2dz

 B.d   3/2
 0 (semi  circle) o I B.d
2   a  z 
2 2
 oI
a 2dz
oI/2
a 2 
2 3/2
2
z 

9. Electrostatic potential of dipole sheet
• Now consider line integral E.dℓ from sheet of electric dipoles
• m = I  I = m/ (density of magnetic moments)
• Replace I by Np (dipole moment density) and o by 1/o
• Contour C is again closed by large semi-circle which
contributes zero to line integral

E.d
Np/2o
E.d  0 (semi  circle) E.d 0
 C

Electric magnetic 
-Np/2o
Field reverses no reversal

10. Differential form of Ampere’s Law
Obtain enclosed current as integral of current density
B.d  I o encl o j.dS
S
B
Apply Stokes’ theorem
j
B.d  B.dS  j.dS
S
o
S
dI  j.dS
dℓ
Integration surface is arbitrary
B o j S
Must be true point wise

11. Ampere’s Law examples
(1) Infinitely long, thin conductor
B is azimuthal, constant on circle of radius r B
o I
B.d o Iencl B 2 r o I  B  2 r
Exercise: find radial profile of B inside and outside conductor
of radius R
B o Ir
Br R 
2 R 2
o I
Br R 
2 r
r
R

12. Solenoid
Distributed-coiled conductor B
Key parameter: n loops/metre
I
If finite length, sum individual loops via B-S Law
If infinite length, apply Ampere’s Law
B constant and axial inside, zero outside
Rectangular path, axial length L
I
L
B vac .d o I encl  B vacL o  nL  I  B vac onI
(use label Bvac to distinguish from core-filled solenoids)
solenoid is to magnetostatics what capacitor is to electrostatics

13. Relative permeability
Recall how field in vacuum capacitor is reduced when
dielectric medium is inserted; always reduction, whether
medium is polar or non-polar:
E vac
E  B rB vac
r is the analogous
expression
when magnetic medium is inserted in the vacuum solenoid.
Complication: the B field can be reduced or increased,
depending on the type of magnetic medium

14. Magnetic vector potential
For an electrostatic field E.d 0 E -
 x E  x  0
We cannot therefore represent B by e.g. the gradient of a scalar
since  x B  o j (rhs not zero)

also .B 0 always (.E  )
o
Magnetostatic field, try B  x A
.B .  x A  0
 x B  x   x A  (see later)
B is unchanged by A'  A  
 x A'  x  A     x A  0

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