Magnetic force 2018b mk

Force between Differential Current Elements 4.. Force between Differential Current Elements 4.. Force between Differential Current Elements 4.. Force between Differential Current Element

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Engineering Electromagnetics

Magnetic Forces & Inductance

Nguy ễ n Công Ph ươ ng

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I Introduction

II Vector Analysis

III Coulomb’s Law & Electric Field Intensity

IV Electric Flux Density, Gauss’ Law & Divergence

V Energy & Potential

VI Current & Conductors

VII Dielectrics & Capacitance

VIII Poisson’s & Laplace’s Equations

IX The Steady Magnetic Field

X Magnetic Forces & Inductance

XI Time – Varying Fields & Maxwell’s Equations

XII Transmission Lines

XIII The Uniform Plane Wave

XIV Plane Wave Reflection & Dispersion

XV Guided Waves & Radiation

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1 Force on a Moving Charge

2 Force on a Differential Current Element

3 Force between Differential Current Elements

4 Force & Torque on a Closed Circuit

5 Magnetization & Permeability

6 Magnetic Boundary Conditions

7 The Magnetic Circuit

8 Potential Energy of Magnetic Fields

9 Inductance & Mutual Inductance

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Force on a Moving Charge (1)

• In an electric field: F = QE

• This force is in the same direction as the EFI (positive charge)

• In a magnetic field: F = Qv B

• This force is perpendicular to both v & B

• In an electromagnetic field: F = Q(E + v B)

• (Lorentz force)

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Ex 2

A test charge Q C, moving with a velocity v = ax + ay m/s, experiences no force in a

region of electric & magnetic fields If the magnetic flux density B = ax – 2azT, find E.

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Ex 3

Given a magnetic flux density B = 10–2 axT, find the

force on an electron whose velocity is 107 m/s:

a) In the x direction, y direction, & z direction.

b) In the xy plane at 45o to the x axis.

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2 Force on a Differential Current Element

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Force on a Differential Current Element (1)

• Force on a differential current element:

dF = dQv B

• If charges are in motion in a conductor, the force is

transferred to the conductor

• Consider only force on conductors carrying currents

• If dQ = ρ v dv (dv is an incremental volume)

→ dF = ρ v dvv B

→ dF = J Bdv

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3 Force between Differential Current Elements

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Force between Differential Current Elements (1)

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Ex 1

Given I1dL1 = – 3ay Am; I2dL2 = – 4az Am

Find the differential force on dL2.

7

2 12

4 10

π π

I1dL1

I2dL2

R12

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I1dL1

I2dL2

R12

Ex 1

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I1dL1

I2dL2

R12

Ex 1

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x

y z

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x

y z

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4 Force & Torque on a Closed Circuit

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Force & Torque on a Closed Circuit (1)

• Force on a filamentary closed circuit:

• If B = const →

• In an electrostatic field:

• → the force on a closed filamentary circuit in a uniform magnetic field is zero

• General: any real closed circuit carrying direct currents

experiences a total vector force of zero in a uniform

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Find the force per meter between two infinite & parallel

filamentary current carrying conductors that are separated d

& carry a current I in opposite direction.

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I0 = 5A, I1 = 3A, I2 = 4A Find the total force on

the wire due to the two loops?

0.6 m 0.8m

0.4 m

c d

e

f

g h

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0.6 m 0.8m

0.4 m

a

b

c d

e

f

g h

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0.6 m 0.8m

0.4 m

a

b

c d

e

f

g h

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1 2 1

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x y

R

B

1

2 3

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• The differential magnetic dipole moment: dm = IdS

• Unit: Am 2

• → dT = dm B

• Holds for differential loops of any shape

• In a uniform magnetic field: T = IS B = m B

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5 Magnetization & Permeability

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Magnetization & Permeability (1)

• The magnetization is defined basing on the magnetic dipole

moment m

• m = I b dS (unit: Am 2 )

• I b : the bound current circulates about a path enclosing dS

• For Δv, the total magnetic dipole moment:

• n: number of magnetic dipole in a unit volume

• Definition of the magnetization:

• M: the (total) magnetic dipole moment per unit volume

0

1

1 lim

n v

i v

∆

=

m total m

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dL

1 lim : the (total) magnetic dipole moment per unit volume

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J

H J

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Magnetization & Permeability (5) I a

ρ

A line current I of infinite extent is within a cylinder of radius a

that has permeability µ , the cylinder is surrounded by free space

Find B, H, & M everywhere, & the current density?

, 0 2

, 2

ϕ

µ

πρ µ

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Magnetization & Permeability (6) I a

ρ

A line current I of infinite extent is within a cylinder of radius a

that has permeability µ , the cylinder is surrounded by free space

Find B, H, & M everywhere, & the current density?

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6 Magnetic Boundary Conditions

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Magnetic Boundary Conditions (1)

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Where z > 0 (region 1), μ = μ1 = 4 μH/m; where z < 0 (region 2), μ2 = 7 μH/m;

at z = 0, given a surface current K = 80axA/m In region 1 there is a magnetic

(2 3 )10

500 750 A/m 4.10

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field B1= 2ax – 3ay + az mT Find B2.

Ex 1

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field B1= 2ax – 3ay + az mT Find B2.

Ex 1

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Air

01000

µ = µ µ = 1000 µ0

B

o30

1 2

A uniform magnetic field of strength B = 1.2 T

exists within an iron core If an air gap is cut

with the orientation shown, find the magnitude

and direction of B in the gap.

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7 The Magnetic Circuit

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The Magnetic Circuit (1)

S

µ

=

d R

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The iron core has an average length of 0.44 m & a

cross-section of 0.02 0.02 m2 The air gap is 2 mm

It is wound with 400 turns Find the current producing

a magnetic flux of 0.141 mWb in the air gap?

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F I

The iron core has an average length of 0.44 m & a

cross-section of 0.02 0.02 m2 The air gap is 2 mm

It is wound with 400 turns Find the current producing

a magnetic flux of 0.141 mWb in the air gap?

3

0.141 10

0.35T (2 10 )(2 10 )

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8 Potential Energy of Magnetic Fields

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Potential Energy of Magnetic Fields (1)

2

1 2 1 2 1 2

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Potential Energy of Magnetic Fields (2)

=

Ex.

Find the magnetic energy associated with unit length of an

infinitely long straight wire of radius a carrying a current I.

I a

2 1

1 2

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9 Inductance & Mutual Inductance

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Inductance & Mutual Inductance (1)

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r

I

N S d

L

µ µ

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0 ln H 2

d b L

a

µ π

→ =

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https://3dwarehouse.sketchup.com/model/ec8884f9 04c69cbb92e83e251d26ee96/Toroidal-Inductor-Coil

b

r

2

NI H

a r

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x y

z

I I

Il d r

r

µ π

−

0

ln 2

l d r L

µ π

− Φ

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2

I I

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• Definition of mutual inductance:

• Φ 12 : flux linking I 1 & I 2

• N 2 : number of turns in circuit 2

• Unit: H

2 12 12

1

N M

I

Φ

=

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Q 1 2

2

Q Q R

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