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Mathematics 2P0

MATH29641, MATH29631

2010-11

Lecture 5

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Week 6 is Reading Week for 2P0/2Q1.

There will be a joint 2P0/2Q1 Lecture on

Monday Nov 1

st

at 4pm in Renold D7!

There will be no 2Q1 Lecture and no

2P0/2Q1 Tutorial on Wednesday Nov 3

rd

!

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2. Vector Calculus

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Vector Calculus

•

To describe mathematically many

electrical or electro-dynamic phenomena

which are observed in experiments the

‘toolbox’ of

vector calculus

is extremely

useful.

2. Vector Calculus

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2. Vector Calculus

2.1. Example Maxwell’s Equations

(i)

Gauss' law

describes the relationship between an

electric field

and the generating

electric charges

:

The electric field tends to point away from positive

charges and towards negative charges. More

technically, it relates the

electric flux

through any

hypothetical

closed

"

Gaussian surface

" to the electric

charge within the surface.

How can this statement be phrased in mathematical terms?

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2. Vector Calculus

2.1. Example Maxwell’s Equations

(ii)

Gauss' law for magnetism

states that there are no

"magnetic charges" (also called

magnetic monopoles

),

analogous to electric charges. Instead the magnetic

field is generated by a configuration called a

dipole

,

which has no magnetic charge but resembles a

positive and negative charge inseparably bound

together. Equivalent technical statements are that the

total

magnetic flux

through any Gaussian surface is

zero, or that the magnetic field is a

solenoidal

vector field

.

How can this statement be phrased in mathematical terms?

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2.1. Example Maxwell’s Equations

(iii)

Faraday's law

describes how a changing

magnetic field

can create ("induce") an

electric field

.

This aspect of

electromagnetic induction

is the

operating principle behind many

electric generators

: A

bar magnet

is rotated to create a changing magnetic

field, which in turn generates an electric field in a

nearby wire.

How can this statement be phrased in mathematical terms?

2. Vector Calculus

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2. Vector Calculus

2.1. Example Maxwell’s Equations

(iv)

Ampère's

law with Maxwell's correction

states that

magnetic fields can be generated in two ways: by

electrical current

(this was the original "Ampère's law")

and by changing electric fields (this was "Maxwell's

correction").

How can this statement be phrased in mathematical terms?

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2. Vector Calculus

2.1. Example Maxwell’s Equations

•

Maxwell’s equations can be formulated very

conveniently using concepts of

vector calculus

.

•

Typically, they can be written either in a

differential

form

using vector differential operators, or in

integral

form

.

•

These differential and Integral forms are related by

famous

vector integral theorems

which will be

discussed here as well.

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2.2.Scalar and Vector Fields

•

A

scalar field

assigns to each point in a

region of a 1D, 2D or 3D domain a scalar

value which is associated with this point.

•

Often this scalar value describes some

physical quantity which (ideally) can be

measured at this point.

•

Even if such a measurement is not done in

practice, a fictitious measurement can be

imagined which would give us this value.

2.2. Scalar and Vector Fields

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2.2.1. Scalar Fields

Examples

for scalar fields are:

•

The temperature at each point in this

lecture theatre.

•

The charge density at each point inside a

cathode ray tube.

•

The pressure at each point in the

atmosphere above Manchester.

•

The total X-ray radiation exposure of each

point in the body during a routine CT.

2.2. Scalar and Vector Fields

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2.2.1. Scalar Fields

Example for a scalar field in 2D:

Each point of the plate is assigned exactly one real

number, which describes the temperature at this point.

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2.2.2. Vector fields

•

A

vector field

is a distribution of vectors over a region

of space such that a vector is associated with each

point of the region.

•

Also these vectors often describe physical quantities

which (ideally) can be measured (sometimes in a

fictitious experiment) at every point.

•

Each of these measurements can be thought of

having a magnitude and a direction which both are

characteristic for each point.

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Example: Coulomb Force

2

1

4

3

2

1

0

2

1

r

r

r

r

q

q

F

=

1

r

Coulomb's law

is a

law

of

physics

describing the

electrostatic interaction between

electrically charged

particles.

In order to obtain both the magnitude and direction of

the force on a charge,

q

1 at position , experiencing a

field due to the presence of another charge,

q

2 at

position , the full

vector

form of Coulomb's law is

2

r

2.2.2. Vector Fields

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2.2. Scalar and vector fields

1

q

2

q

A graphical representation of Coulomb's law. You

get a vector field if you keep fixed and move

around and verify the Coulomb force at each

location of .

2

q

2.2.2. Vector Fields

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A visualization of the Coulomb force field could look like this:

2.2.2. Vector Fields

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Other examples for vector fields are

•

The water velocity at each point in a river

2.2.2. Vector Fields

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•

The gravitational pull of the earth

2.2.2. Vector Fields

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•

If the value (vector) assigned to a given

point does not depend on the time when a

(fictitious) measurement is made, the field

is said to be

stationary

.

•

If the value (vector) changes with time,

then the field is said to be

time-dependent

.

2.2. Scalar and Vector Fields

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•

In the following we will study the properties

of scalar and vector fields, in particular

arising from

electromagnetic phenomena

.

•

However, the

mathematical concepts

used

in our description are

far more general

, and

can be used as well for describing different

physical phenomena such as

Fluid Flow

,

Elasticity

,

Traffic Control

,

Molecular

Dynamics

, …

2.2. Scalar and Vector Fields

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Further Reading:

•

HELM modules 28.1 and 28.2.

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