Magnetostatics Short notes | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

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Short Notes on Magnetostatics
1. In tro duction
• Magnetostatics studies magnetic fields pro duced b y steady (time-in v arian t) curren ts and p ermanen t
magnets.
• Analogous to electrostatics, it assumes no time-v arying fields, simplifying Maxw ell’s equations.
• Key quan tities: Magnetic field B , magnetic field in tensit y H , curren t densit yJ .
2. F undamen tal La ws
• Biot-Sa v art La w : Magnetic field dB due to a curren t elemen t Idl at p ositionr :
dB =
µ
0
I
4p
dl׈ r
r
2
, B =
µ
0
I
4p
Z
dl׈ r
r
2
where µ
0
=4p×10
-7
H/m is the p ermeabilit y of free space.
• Amp ere’s La w : Relates magnetic field to curren t enclosed b y a lo op:
I
C
B·dl =µ
0
I
enc
, ?×B =µ
0
J
• Gauss’s La w for Magnetism : Magnetic field is div ergence-free (no magnetic monop oles):
?·B =0
3. Magnetic Field and Material Prop erties
• Magnetic Field In tensit y : H =
B
µ0
-M in materials, whereM is magnetization.
• Magnetic P ermeabilit y : B =µH , where µ =µ
r
µ
0
and µ
r
is relativ e p ermeabilit y .
• Materials :
– Diamagnetic: µ
r
˜ 1 , w eak opp osition toB .
– P aramagnetic: µ
r
> 1 , w eak alignmen t withB .
– F erromagnetic: µ
r
» 1 , strong magnetization (e.g., iron).
4. Magnetic V ector P oten tial
• Defined as A , whereB =?×A , ensuring?·B =0 .
• F or a curren t distribution:
A =
µ
0
4p
Z
ldl
'
|r-r
'
|
• Simplifies calculations for B in comple x geometries.
5. Boundary Conditions
• Normal comp onen t ofB is con tin uous: B
1n
=B
2n
.
1
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Short Notes on Magnetostatics
1. In tro duction
• Magnetostatics studies magnetic fields pro duced b y steady (time-in v arian t) curren ts and p ermanen t
magnets.
• Analogous to electrostatics, it assumes no time-v arying fields, simplifying Maxw ell’s equations.
• Key quan tities: Magnetic field B , magnetic field in tensit y H , curren t densit yJ .
2. F undamen tal La ws
• Biot-Sa v art La w : Magnetic field dB due to a curren t elemen t Idl at p ositionr :
dB =
µ
0
I
4p
dl׈ r
r
2
, B =
µ
0
I
4p
Z
dl׈ r
r
2
where µ
0
=4p×10
-7
H/m is the p ermeabilit y of free space.
• Amp ere’s La w : Relates magnetic field to curren t enclosed b y a lo op:
I
C
B·dl =µ
0
I
enc
, ?×B =µ
0
J
• Gauss’s La w for Magnetism : Magnetic field is div ergence-free (no magnetic monop oles):
?·B =0
3. Magnetic Field and Material Prop erties
• Magnetic Field In tensit y : H =
B
µ0
-M in materials, whereM is magnetization.
• Magnetic P ermeabilit y : B =µH , where µ =µ
r
µ
0
and µ
r
is relativ e p ermeabilit y .
• Materials :
– Diamagnetic: µ
r
˜ 1 , w eak opp osition toB .
– P aramagnetic: µ
r
> 1 , w eak alignmen t withB .
– F erromagnetic: µ
r
» 1 , strong magnetization (e.g., iron).
4. Magnetic V ector P oten tial
• Defined as A , whereB =?×A , ensuring?·B =0 .
• F or a curren t distribution:
A =
µ
0
4p
Z
ldl
'
|r-r
'
|
• Simplifies calculations for B in comple x geometries.
5. Boundary Conditions
• Normal comp onen t ofB is con tin uous: B
1n
=B
2n
.
1
• T angen tial comp onen t ofH is con tin uous in the absence of surface curren ts:
H
1t
=H
2t
, or H
1
׈ n-H
2
׈ n =K
whereK is surface curren t densit y .
6. Applications
• Design of electromagnets, inductors, and transformers.
• Magnetic shielding in electronic devices.
• Magnetic field analysis in MRI mac hines.
• Motors and generators in electrical engineering.
7. Practical Considerations
• Symmetry : Use Amp ère’s la w for symmetric curren t distributions (e.g., infinite wire, solenoid).
• Numerical Metho ds : Required for complex geometries where analytical solutions are impracti-
cal.
• Material Nonlinearit y : F erromagnetic materials exhibit h ysteresis, complicating B-H rela-
tionship.
• Field P enetration : High µ
r
materials concen trate magnetic flux, affecting field distribution.
8. Conclusion
• Magnetostatics pro vides a framew ork for analyzing steady-state magnetic fields using Biot-Sa v art,
Amp ère’s, and Gauss’s la ws.
• Understanding v ector p oten tial, b oundary conditions, and material prop erties is crucial for practical
applications in electromagnetic devices.
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