Download Biot Savart law Ampere`s circuital law Faradays laws of

 Biot
Savart law
Ampere’s circuital
law
Faradays laws of
Electromagnetic
induction
Electromagnetic
waves,
Divergence, Curl
and Gradient
Maxwell’s
Equations
1
Biot
- Savart law is used to calculate the magnetic field
due to a current carrying conductor.
According to this law, the magnitude of the magnetic
field at any point P due to a small current element I.dl (
I = current through the element, dl = length of the
element) is,
Y
Idl sin θ
dB ∝
r2
µ0 Idlsinθ
dB = .
2
4π
r
I
B
I.dl C
A
In vector
notation,
µ
idl × r
dB =
.
4π
r3
0
θ
r
dB
P
X
2
It
states that the line integral of the
magnetic field (vector B) around any closed
path or circuit is equal to µ0 (permeability of
free space) times the total current (I) flowing
through the closed circuit.
Mathematically,
→ →
∫ B . dl = µ 0 I
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Michael Faraday found that whenever there is a
change in magnetic flux linked with a circuit, an emf
is induced resulting a flow of current in the circuit.
The magnitude of the induced emf is directly
proportional to the rate of change of magnetic flux.
Lenz’s rule gives the direction of the induced emf
which states that the induced current produced in a
circuit always in such a direction that it opposes the
change or the cause that produces it.
dφ
induced emf (e ) = −
dt
dφ is the change magnetic flux linked with a circuit
4
According
to Maxwell’s modification of
Ampere’s law, a changing electric field gives
rise to a magnetic field.
It leads to the generation of electromagnetic
disturbance comprising of time varying
electric and magnetic fields.
These disturbances can be propagated
through space even in the absence of any
material medium.
These disturbances have the properties of a
wave and are called electromagnetic waves.
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If the (average) velocity of the charge carriers is constant
through the material distribution and the B-field is also
constant:
◦ Reasonable assumption for steady currents
The charge distribution can also be affected by an electric field
Can’t be applied (yet
yet)
yet to moving point particles
◦ Ampere/Biot-Savart’s law is not applicable to point charges
◦ The charge will radiate energy and the system won’t satisfy curl(E
E)=0
◦ Nevertheless, we will eventually reach the conclusion that Lorentz’
force holds for any charges in motion
We often ignore these non-steady state effects in semi-realistic
problems
E.g.: Cyclotron, cycloid motion, etc.
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Similarly as what was done in electrostatics, we would like
to use the definition of work and potential energy in
mechanics together with the concept of magnetic force
from the Ampere’s law to define the work of a magnetic
force and the energy of a magnetic field
But… the magnetic force is always perpendicular to the
direction of the flow of charge
The magnetic force may alter the direction in which a
charged
system moves, but cannot speed it up or slow it down
A magnetic force does NO work on a current
This doesn’t mean that there is no energy stored in a
magnetic field. It means that we will need to proceed
differently than what we did in electrostatics to define such
magnetic energy
◦ Need to have electrodynamics and Faraday’s law to define a
procedure to determine the energy stored in a magnetic field
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