Transcript Slide 1

Lecture 8
o Aim of the lecture
 Calculation of Magnetic Fields
 Biot-Savart Law
 Magnetic field, B
 Ampres Law
 B field
 Case of static E field
o Main learning outcomes
 familiarity with
 Calculation of Magnetic Fields from currents
 Biot-Savart Law
 Amperes Law
o We understand:
o the source of electric field is the electric charges,
o But there are no magnetic monopoles,
o which means no magnetic charges
o So where do magnetic fields come from?
The answer is that it must be remembered
There is a VERY close connection between
electric and magnetic fields
A
magnetic
field is just
an electric
fieldfield,
in motion
( loosely
speaking
– there
is just one
it is
or
an electromagnetic field
An
electric field
just a magnetic
fieldwhen
in motion
It becomes
pureismagnetic
or electric
the
charge is stationary)
o Hence it is possible to have a magnetic field when there
are no magnetic charges.
o What is needed is electric charges in motion
 This is a CURRENT
However:
o The shape of the field that can be made is not the same
 A stationary electric charge makes a monopole electric field
 Two opposite electric charges makes an electric dipole field
 A string of electric charges moving in a loop
(a current loop) makes a dipole magnetic field
 There is no monopole magnetic field observed in nature
simply because there are no magnetic monopoles seen
Monopole electric field (electric charge)
(remember this cannot be exactly
correct in a 2-D drawing)
Dip0le
electric field
field (electric
(electric charges)
charges)
Dipole
magnetic
Same general field pattern
using a single current loop
DIPOLE
More detail on how to compute Magnetic Fields caused by currents
ds
o The Biot-Savart Law
 tells for each current element, Ids
 the size and direction of the B field produced
s
o The total field is then the sum of all such elements.
Produces a B field element
of constant magnitude on
this ring
This current element
ds
Also a constant
magnitude on this
ring
The element contributes everywhere in space according to the
Biot-Savart Formula
To get the total field here
ds
Need to add all the
contributions from all the current
elements
Vector Integral needed
B =
∫
∫
Note: in principle
o All currents in universe
o But because of 1/r2
o Only nearby matter
Biot
Savart
What this says is simply that integrating the B field round any closed loop
will give you an answer which is proportional to the current flowing
through the loop.
It doesn't matter what shape the loop is
(this is the case where the electric field is constant – see next slide)
The complete version is this:
Ampère
This equation relates the
magnetic field to the motion
of charges and the change in
electric field.
In many cases the electric field
is constant, so that the second
term can be dropped.
J is current density
This is the case for fixed current
distributions, such as currents in
wires.
( which we did on the previous
transparency )
So this is Ampere's Law
in differential form
It is telling us how the
magnetic and electric
fields are related to the
electric charges.
(You are not expected to manipulate Maxwell’s equations in this course)
Next Lecture:
Faraday’s Law of Induction