b-field of a rotating charged conducting sphere1 magnetic field of a rotating charged conducting...

B-field of a rotating charged conducting sphere

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Magnetic Field of a Rotating Charged Conducting SphereMagnetic Field of a Rotating Charged Conducting Sphere

© Frits F.M. de Mul


2



Question:

Calculate B-field in arbitrary points on the axis of rotation inside and outside the sphere

Question:

Calculate B-field in arbitrary points on the axis of rotation inside and outside the sphere

Available:

A charged conducting sphere (charge Q, radius R), rotating with rad/sec

Available:

A charged conducting sphere (charge Q, radius R), rotating with rad/sec


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Calculate B-field in point P inside or outside the sphere

Calculate B-field in point P inside or outside the sphere

P

P

O

Analysis and Symmetry (1)Analysis and Symmetry (1)

Assume Z-axis through O and P.Assume Z-axis through O and P.

zP

Z

Y

XCoordinate systems:

- X,Y, Z

Coordinate systems:

- X,Y, Z

r

- r, - r,


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Conducting sphere,

all charges at surface:

surface density: Q/(4R2) [C/m2]

Conducting sphere,

all charges at surface:

surface density: Q/(4R2) [C/m2]

P

P

zP

Y

X

Z

r

ORotating charges will establish a “surface current”

Rotating charges will establish a “surface current”

Surface current density j’ [A/m] will be a function of

Surface current density j’ [A/m] will be a function of

j’


5


P

zP

Y

X

Z

r

O

T

Cylinder- symmetry around Z-axis:

Cylinder- symmetry around Z-axis:

dBz

Z-components only !!


Direction of contributions dB:Direction of contributions dB:

P

O

dB

T

r

er

dl

20 .

4 Pr

I redldB

20 .

4 Pr

I redldB

Biot & Savart :Biot & Savart :

rPdB dB, dl and er mutual. perpendic.dB, dl and er mutual. perpendic.


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Approach (1): a long wireApproach (1): a long wire

dB

20 .

4 Pr

I redldB

20 .

4 Pr

I redldB


note:

r and vector er !!

note:

r and vector er !!

dB dl and erdB dl and er

dB AOPdB AOP

Z

YX

P

z

I.dl in dz at zI.dl in dz at z

dl

er rP

yP

A

O


7

Approach (2): a volume currentApproach (2): a volume current

dB

dvrP

20

4rej

dB

dvrP

20

4rej

dB



dB AOPdB AOP

j: current density [A/m2]

j: current density [A/m2]

Z

Y

P

j.dA.dl = j.dvj.dA.dl = j.dvdl

er

yP

dA

jA

OrP


8

Approach (3): a surface currentApproach (3): a surface current

dB

dArP

20

4rej

dB

dArP

20

4rej

dB



dB AOPdB AOP

Z

Y

P

dl

er

yP

dl

j’A

OrP

Current strip at surface: j’: current density[A/m]

Current strip at surface: j’: current density[A/m]j’.db.dl = j’.dAj’.db.dl = j’.dA

dldb


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Approach (4)Approach (4)

Z

d

R

d

R sin

Conducting sphere,

surface density: Q/(4R2)

Conducting sphere,

surface density: Q/(4R2)

surface element:

dA = (R.dR.sind

surface element:

dA = (R.dR.sind

R.d.R.sind

Surface element:


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Conducting sphere (1) Conducting sphere (1)

dA = db.dl dA = db.dl

Surface charge.dAon dA will rotate with

Surface charge.dAon dA will rotate with

dl = R.sinddb= R d

dl = R.sinddb= R d Needed:

• j , er , rP

Needed:

• j , er , rP

dArP

20

4rej

dB

dArP

20

4rej

dB

with j’ in [A/m]with j’ in [A/m]

R.sind

Z

R

d

d

R sin R.d


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Z

R

d

d

R sin

R.d R.sind

dA = db.dl dA = db.dl

dl = R.sinddb= Rd

dl = R.sinddb= Rd

Full rotation over 2Rsinin 2 s.

Full rotation over 2Rsinin 2 s.

Charge on ring with radius R.sin and width db is: . 2R.sindb

Charge on ring with radius R.sin and width db is: . 2R.sindb

current: dI = .2R.sindb / (2) = R sindb

current: dI = .2R.sindb / (2) = R sindb

current density: j’ =R sin [A/m]

current density: j’ =R sin [A/m]


12


R

d

d

R sin

R.d R.sind

dArP

20

4rej

dB

dArP

20

4rej

dB

P

zP

j’

errP

dA = R.d. R.sinddA = R.d. R.sind

j’ er :

=> | j’ x er | = j’.er = j’

j’ er :

=> | j’ x er | = j’.er = j’

j’ = R sin j’ = R sin


13


R

d

d

R sin

P

zP

j’

errP

dArP

20

4rej

dB

dArP

20

4rej

dB

dA = Rd R.sinddA = Rd R.sind



dBz

Pr

R sincos

Pr

R sincos

Cylinder- symmetry:

Cylinder- symmetry: P

O

dB

R

rPzP er



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R

d

d

R sin

P

zP

j’

errP

dArP

20

4rej

dB

dArP

20

4rej

dB

dA = Rd.R.sind

dA = Rd.R.sind

P

O

dBdBz

R

rPzP

Pr

R sincos

Pr

R sincos

rP2= (R.sin)2 +

(zP - R.cos)2

rP2= (R.sin)2 +

(zP - R.cos)2

PP

z r

RdRdR

r

RdB

sinsin...

sin

4 20

PP

z r

RdRdR

r

RdB

sinsin...

sin

4 20



15


R

d

d

R sin

P

zP

j’

errP

with rP2= (R.sin)2 + (zP - R.cos)2with rP

2= (R.sin)2 + (zP - R.cos)2

dd

r

RdB

P

z .sin

4 3

340

ddr

RdB

P

z .sin

4 3

340

Integration: 0<<

Integration: 0<<

PP

z r

RdRdR

r

RdB

sinsin...

sin

4 20

PP

z r

RdRdR

r

RdB

sinsin...

sin

4 20


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P

P

zP

Y

X

Z

R

O

this result holds for zP>R ;

for -R<zP<R the result is:

zeB R03

2 zeB R03

2

and for zP<-R: zeB 3

40

.3

2

pz

R

zeB 3

40

.3

2

pz

R

zeB 3

40

.3

2 :result

pz

R zeB 3

40

.3

2 :result

pz

R


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inside sphere: constant field !! inside sphere: constant field !!

P

P

zP

Y

X

Z

r

O

result for |zP|>R :result for |zP|>R :

result for |zP|<R :result for |zP|<R :

zeB 3

40

.3

2

pz

R zeB 3

40

.3

2

pz

R

zeB R03

2 zeB R03

2

B directed along +ez for all points everywhere on Z-axis !!

B directed along +ez for all points everywhere on Z-axis !!


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With surface density: Q/(4R2) :

result for |zP| > R :

result for |zP| > R : zz eeB 3

20

3

40

6.3

2

pp z

RQ

z

R

zz eeB 3

20

3

40

6.3

2

pp z

RQ

z

R

result for |zP| < R :

result for |zP| < R : zz eeB

R

QR

63

2 0

0 zz eeBR

QR

63

2 0

0


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Plot of B for:

Q = 1

0 = 1

= 1

(in SI-units)

Plot of B for:

Q = 1

0 = 1

= 1

(in SI-units)

zP / R

zeB 3

20

6

pz

RQ

zeB 3

20

6

pz

RQ

zeBR

Q

6 0 zeB

R

Q

6 0


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Conclusions (1)Conclusions (1)

Homogeneously charged sphere

(see other presentation)


(see other presentation)

zeB3

20

10

Pz

RQ

zeB3

20

10

Pz

RQ

zeB 223

0 3520

PzRR

Q

zeB 223

0 3520

PzRR

Q

|zP| < R|zP| > R

Conducting sphereConducting sphere

|zP| > R |zP| < R

zeB3

20

6

pz

RQ

zeB3

20

6

pz

RQ

zeB

R

Q

6 0 zeB

R

Q

6 0


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Conclusions (2)Conclusions (2)

Plot of B for:

Q = 1

0 = 1

= 1

(in SI-units)

Plot of B for:

Q = 1

0 = 1

= 1

(in SI-units)

zP / R



Conducting sphereConducting sphere

The end !!The end !!

b-field of a rotating charged conducting sphere1 magnetic field of a rotating charged conducting...

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