Formula Sheets: Magnetostatics | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Magnetostatics (EMFT) – GATE
1. Basic Concepts
• Magnetostatics: Study of steady magnetic ?elds produced by constant currents.
• Constants:
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Magnetic ?eld: B (unit: Tesla), Magnetic ?eld intensity: H (unit: A/m).
• Relation: B =µ 0
H (in free space).
2. Biot-Savart Law
• Magnetic Field due to Current Element:
dB =
µ 0
4p
Idl׈ r
r
2
where I: Current, dl: Di?erential length, r: Distance, ˆ r: Unit vector.
• Total Field:
B =
µ 0
I
4p
Z
dl׈ r
r
2
3. Magnetic Field Examples
• In?nite Straight Wire:
B =
µ 0
I
2pr
(at distance r, direction by right-hand rule).
• Circular Loop (Center):
B =
µ 0
I
2R
where R: Radius.
• Solenoid (Inside):
B =µ 0
nI
where n =
N
L
: Turns per unit length.
4. Amperes Circuital Law
• Integral Form:
I
C
H·dl =I
enc
where I
enc
: Total current enclosed by path C.
• Di?erential Form:
?×H =J
whereJ: Current density (A/mš).
1
Page 2


Formula Sheet for Magnetostatics (EMFT) – GATE
1. Basic Concepts
• Magnetostatics: Study of steady magnetic ?elds produced by constant currents.
• Constants:
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Magnetic ?eld: B (unit: Tesla), Magnetic ?eld intensity: H (unit: A/m).
• Relation: B =µ 0
H (in free space).
2. Biot-Savart Law
• Magnetic Field due to Current Element:
dB =
µ 0
4p
Idl׈ r
r
2
where I: Current, dl: Di?erential length, r: Distance, ˆ r: Unit vector.
• Total Field:
B =
µ 0
I
4p
Z
dl׈ r
r
2
3. Magnetic Field Examples
• In?nite Straight Wire:
B =
µ 0
I
2pr
(at distance r, direction by right-hand rule).
• Circular Loop (Center):
B =
µ 0
I
2R
where R: Radius.
• Solenoid (Inside):
B =µ 0
nI
where n =
N
L
: Turns per unit length.
4. Amperes Circuital Law
• Integral Form:
I
C
H·dl =I
enc
where I
enc
: Total current enclosed by path C.
• Di?erential Form:
?×H =J
whereJ: Current density (A/mš).
1
5. Magnetic Vector Potential
• De?nition: B =?×A.
• Potential due to Current:
A =
µ 0
4p
Z
Idl
r
• Coulomb Gauge: ?·A = 0.
6. Magnetic Force
• Force on a Moving Charge (Lorentz Force):
F =q(v×B)
where q: Charge,v: Velocity.
• Force on Current-Carrying Conductor:
F =I(l×B)
wherel: Length vector of conductor.
7. Magnetic Flux
• Magnetic Flux:
F =
ZZ
S
B·dS
• Gausss Law for Magnetism:
I
S
B·dS = 0
• Di?erential Form:
?·B = 0
8. Inductance
• Self-Inductance:
L =
F
I
• Mutual Inductance:
M =
F
21
I
1
where F
21
: Flux in coil 2 due to current I
1
in coil 1.
• Solenoid Inductance:
L =µ 0
n
2
Al
where A: Cross-sectional area, l: Length.
2
Page 3


Formula Sheet for Magnetostatics (EMFT) – GATE
1. Basic Concepts
• Magnetostatics: Study of steady magnetic ?elds produced by constant currents.
• Constants:
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Magnetic ?eld: B (unit: Tesla), Magnetic ?eld intensity: H (unit: A/m).
• Relation: B =µ 0
H (in free space).
2. Biot-Savart Law
• Magnetic Field due to Current Element:
dB =
µ 0
4p
Idl׈ r
r
2
where I: Current, dl: Di?erential length, r: Distance, ˆ r: Unit vector.
• Total Field:
B =
µ 0
I
4p
Z
dl׈ r
r
2
3. Magnetic Field Examples
• In?nite Straight Wire:
B =
µ 0
I
2pr
(at distance r, direction by right-hand rule).
• Circular Loop (Center):
B =
µ 0
I
2R
where R: Radius.
• Solenoid (Inside):
B =µ 0
nI
where n =
N
L
: Turns per unit length.
4. Amperes Circuital Law
• Integral Form:
I
C
H·dl =I
enc
where I
enc
: Total current enclosed by path C.
• Di?erential Form:
?×H =J
whereJ: Current density (A/mš).
1
5. Magnetic Vector Potential
• De?nition: B =?×A.
• Potential due to Current:
A =
µ 0
4p
Z
Idl
r
• Coulomb Gauge: ?·A = 0.
6. Magnetic Force
• Force on a Moving Charge (Lorentz Force):
F =q(v×B)
where q: Charge,v: Velocity.
• Force on Current-Carrying Conductor:
F =I(l×B)
wherel: Length vector of conductor.
7. Magnetic Flux
• Magnetic Flux:
F =
ZZ
S
B·dS
• Gausss Law for Magnetism:
I
S
B·dS = 0
• Di?erential Form:
?·B = 0
8. Inductance
• Self-Inductance:
L =
F
I
• Mutual Inductance:
M =
F
21
I
1
where F
21
: Flux in coil 2 due to current I
1
in coil 1.
• Solenoid Inductance:
L =µ 0
n
2
Al
where A: Cross-sectional area, l: Length.
2
9. Magnetic Materials
• Magnetization: M =?
m
H, where ?
m
: Magnetic susceptibility.
• Relation:
B =µ 0
(H+M) =µ H, µ =µ 0
(1+?
m
)
• Boundary Conditions:
H
tangential
continuous, B
normal
continuous
10. Energy in Magnetic Fields
• Energy Stored in Inductor:
W =
1
2
LI
2
• Magnetic Energy Density:
w =
1
2
B·H =
B
2
2µ 0
11. Design Considerations
• Symmetry: Use Amperes law for cylindrical, solenoidal, or planar symmetry.
• Boundary Conditions: Apply at interfaces between magnetic materials.
• Applications: Inductors, transformers, magnetic shielding, motor design.
• Numerical Methods: Solve for complex geometries using Biot-Savart or vector
potential.
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