**
Field Theory - Online Tutorial 3 **

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

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 = ε_{o}
and conductivity constant = σ

Find the following...

1a) Write down Laplace's Equation

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

1c) **E**

1d)
ρ_{sa} 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 | B | C | D |

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__SOLUTION 1a)__

Yes, very simple ... its **∇ ^{2}V
= 0**

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

**∇ ^{2}V
= −
ρ_{v}/ ε** .... Poisson's equation

when you are in charge-free region,
which
ρ_{v} = 0

so it became.. **∇ ^{2}V
= 0**

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

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

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

A |
B |
C |

**
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 |
B |
C |

**
SOLUTION 1c)**

Simply **E** = − ∇*V*

**1d)** Find
ρ_{sa} at r
= a (where ε = ε_{o})

Suggest Solving Time: 5 min

A |
B |
C |

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**
SOLUTION 1d)**

* If you choice A ...
ρ_{sa} 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 *w _{e}*

**1h)** Find Total Energy

Suggest Solving Time: 3 min

**
SOLUTION 1h)**

The symbol of Total Energy is capital case W_{e}

* 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 | B |

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 :)

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