**
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:**

For the possible Final Exam Question, such as may ask you to

1) write down the Maxwell Equations

2) similar question using Ampere's Law or Bio Savart Law, but the shape of object change to semicircle

3)
Question asking for V_{emf} is 99.99% on every year's final exam, please
don't forget to review it (it was on last tutorial).

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 :)