Field Theory - Online Tutorial 3

by Kevin Tang , last update: Jul 1st, 2014

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Q1 -

Given a spherical capacitor that have a inner and outer radius a [m] and b [m].

Where voltage will be following... V(r=a) = 5 [V], V(r=b) = 0[V] as shown in Figure #1.  Given ε = εo and conductivity constant = σ

Find the following...

 

1a) Write down Laplace's Equation

1b) V (a < r < b)

1c) E

1d) ρsa when r = a

1e) Q when r= a

1f) C at r =a

1g) Energy Density

1h) Total Energy

1i) G conductivity, when r = a, given conductivity constant = σ

 

                                Figure # 1

 

 

1a) Write down Laplace's Equation

 

Solving Time : 10 sec

If you are not sure which one is Laplace's Equation, please review it right now~

 

   
A
  • ∇V = 0
  • Wrong Answer
B
  • ∇E = 0
  • Wrong Answer
C
  • 2V = 0
  • Correct!
D
  • 2E = 0
  • Wrong Answer

 

 

 

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SOLUTION 1a)

Yes, very simple ... its 2V = 0

 

Notice, Laplace's equation is from the Poisson's equation, its a "special case" of Poisson equation, where as following...

 

2V = − ρv/ ε  .... Poisson's equation

when you are in charge-free region, which ρv = 0

so it became.. 2V = 0

 

 

1b) Find V (a<r<b)

 

Solving Time: 10 min   ... lets give a try :D

 

 

V (a < r < b) = ? which one?

 

A

  • (5/(1/b − 1/a))(1/b −1/r) [V]
  • Correct!

 

B

  • (5/(1/a − 1/b)(1/a − 1/r)[V]
  • Wrong Answer

 

C

  • (1/(1/b − 1/a))(1/a − 1/r)[V]
  • Wrong Answer

 

 

 

SOLUTION 1b)

 

Here is how you solve it, please actually go over it at least "once" by yourself.

 

 

(Please don't forget the condition, that r CANNOT equal to zero)

 

 

Next, our target is to find A and B, 2 unknown so you need at least 2 equations,

sub the voltage value, V(r=a) = 5 and V(r= b) = 0 to make 2 equations.

 

 

Now, let's find out A = ? and  B = ?

 

...

 

Almost finished...

 

 

The answer for V(a<r<b)

 

 

....

1c) Find E

 

Maximum solving time: 5 min

 

A

  • −1/r2 (5/(1/a − 1/b)) ar [V/m]
  • Wrong Answer

 

B

  • 1/r2 (5/(1/a − 1/b)) ar [V/m]
  • Correct!

 

C

  • 1/a2 (5/(1/a − 1/b)) ar [V/m]
  • Wrong Answer

 

 

SOLUTION 1c)

 

Simply E = −V

 

 

1d) Find ρsa at r = a  (where ε = εo)

 

Suggest Solving Time: 5 min

 

A

  • εo/a2 (5/(1/a − 1/b)) ar [C/m2]
  • Wrong Answer

 

B

  • εo/r2 (5/(1/a − 1/b)) [C/m2]
  • Wrong Answer

 

C

  • εo/a2 (5/(1/a − 1/b)) [C/m2]
  • Correct!

 

 

 

 

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SOLUTION 1d)

 

 

 

* If you choice A ... ρsa is a scalar not a vector.

* If you choice B ... don't forget to sub r = a at the end.

* If you choice C ... you are cool :D

 

 

1e) Find Q when r= a

 

Suggest Solving Time: 2 min

 

 

 

SOLUTION 1e)

 

 

1f) Find C when r = a

 

Suggest Solving Time: 1 min

 

 

 

SOLUTION 1f)

 

1g) Find Energy Density

 

Suggest Solving Time: 2 min

 

 

 

 

SOLUTION 1g)

 

The symbol of energy density is small case we

 

 

1h) Find Total Energy

 

Suggest Solving Time: 3 min

 

 

SOLUTION 1h)

 

The symbol of Total Energy is capital case We

* Note: The integral of surface area respect to the change of radius, can lead you directly to the volume, no need to perform triple integral here.

 

 

 

 

1i) G conductivity, when r = a, given conductivity constant = σ

 

Suggest Solving Time: 3 min

 

 

 

 

SOLUTION 1i)

 

 

 

 

Q2

An infinitely long transmission line consisting of two concentric cylinders having their axes along z-axis.  The inner conductor has radius "a"

and carrying current I, while the outer conductor has inner radius b and thickness t and carries return current - I.

 

Find the following...

 

a) Draw the diagram (show the detail label)

b) Find the magnetic field strength H everywhere

c) Find the magnetic flux B everywhere

d) Find the self inductance of inner conductor

 

SOLUTION a)

 

 
A
  • a)
  • Wrong Answer
B
  • b)
  • Correct Answer

 

If you choice A, you got it wrong, as it said the inner conductor has radius of "a", the current is running inside it, so the i = +I will be inside "a"

If you choice B, you are cool~ :D

 

b) Find the magnetic field strength H everywhere

 

SOLUTION b)

First you should able to see there are "4" regions.

Region 1 :  0 < ρ < a

Region 2 :  a < ρ < b

Region 3 :  b < ρ < b + t

Region 4 :  b+ t < ρ

 

Remember, left side stay the same, only the right hand side is chaging

 

For Region 1





For Region 2






For Region 3
Notice, the LS side is still the same (line integral part)






For Region 4
H = 0 [A/m].... why?

Due to the the total enclosed current equal to zero when you are outside b+t, as I + (-I) = 0

 

 

 

c) Find

 

d) Self Inductance

 

Last note:

 

 

Good luck, its my very pleasure to work with all of you :)

 

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You may leave me any message if you have any questions, I will try to answer ASAP, thanks and good luck :)