PPT No. 18
* Ampère’s circuital law
* Applications of Ampère’s circuital law
* Ampère
Ampère's Circuital Law
Ampère's Circuital law relates
the magnetic B field to
its source, the current density J.
Statement of the Law :
The line integral of magnetic B-field
around any closed path or loop is equal to
the Permeability times the net electric current
through the area enclosed by the loop.
Ampère's Circuital Law
For infinitely narrow wire passing current I
The line integral of magnetic induction B
around a closed curve (or loop) is equal to
μ0 times the net current I
through the area bounded by the curve
Ampère's Circuital Law
For wire passing current density J
in finite cross-section area S
The line integral of magnetic induction B
around a closed curve (or loop) is equal to
μ0 times the surface integral of
the dot product of
Current density j and area element dS
Ampère's Circuital Law
Consider an infinitely long straight conducting wire
carrying a steady current I in free space.
Its magnetic B field can be represented by
The magnetic field lines in form of
concentric circular loops surrounding it
Consider a circular Amperian loop in the form of
a closed circular loop of radius r,
with centre on the conductor
in the plane perpendicular to the conductor.
Magnetic Field of A Straight Conducting Wire
Fig. Amperian loop of a straight, Current carrying wire
Magnetic Field of A Straight Conducting Wire
Constant B-field on the perimeter of Amperian loop
Ampère's Circuital Law
Every point on the Amperian loop is
at the same distance from the conductor.
The total length of the path is equal to
the circumference of the circle: ℓ = 2 π r
The magnitude of magnetic B- field
at any point on the loop
is given by Biot Savart law
B = μ0 I / 2 π r
Ampère's Circuital Law
Line integral of magnetic induction B
around this closed loop is
= 2πrB
= μ0 I
This equation is called as the Ampere's Law.
Ampère's Circuital Law
In general,
when the magnetic field B is always directed
along the tangent to the perimeter of a closed curve and
has constant value B at all points on the perimeter then
B field x Perimeter of closed loop
= μ0 x Total current passing through
the surface enclosed by the loop
Ampère's Circuital Law
The equation for the line integral of B is
independent of the position relative to the source.
It is valid for any closed loop around the wire,
the path need not be circular or perpendicular to the wires.
It is true for distorted paths or
Irregular closed loops or
complicated magnetic fields.
The circulation of B over a closed loop
which does not enclose the current carrying conductor
is zero as the current in enclosing loop is zero
Ampère's Circuital Law
Equation for Ampère's Circuital Law
can be generalized for any number of wires
carrying any number of currents in any direction.
The resultant magnetic field B is due to the net current
(algebraic sum = I1 - I2 - I3 + I4)
through the surface enclosed by the loop around the wires
This is the Principle of Superposition
Principle of Superposition
The resultant magnetic field B is due to many
current carrying wires enclosed by a closed path
Ampère's Circuital Law
Ampere’s law in Magnetostatics is analogues to
the Gauss law in electrostatics in the sense that
Ampere’s law connects the line integral of magnetic B-field
to enclosed electric currents, while
Gauss’ law connects the surface integral of electric field
to enclosed charge density
Ampere’s law also has differential and integral forms
which are equivalent.
Ampère's Circuital Law
Ampere’s law and Biot Savart’s law
can be derived from each other and
are equivalent in scientific conent.
However, in the case of Biot Savert law,
evaluation of complicated integrals is required.
It can be avoided by using Ampere’s law.
Its power is evident in simplifying physical situations
where B-field has some kind of symmetry
(e.g. circular, cylindrical etc. symmetry)
Applications of Ampere’s Law
Magnetic Field of a Solenoid
A solenoid is a long wire carrying steady current
wound closely in the form of a helix.
The wire is coated with insulating material
so the adjacent turns are electrically insulated.
A solenoid carrying current I, having radius a, length L,
Total number of turns N, Number of turns/unit length n = N/L
has magnetic field B at a point P well inside the solenoid
B = μ0nI
B is independent of the length and diameter of the solenoid
& uniform over the cross-section of the solenoid
Applications of Ampere’s Law
Magnetic Field of a Solenoid
Applications of Ampere’s Law
Toroid:
The magnetic field within the volume enclosed by a toroid
consisting of N evenly spaced turns of wire carrying current I
at a point at distance R from the axis passing through centre
and perpendicular to the plane of toroid is given by
B is not constant at different points
over the cross-section of toroid (unlike solenoid).
Applications of Ampere’s Law
Magnetic Field of a Toroid
The magnetic field within the volume enclosed by a toroid
Ampère
The ampere is the SI unit of Electric current and
is one of the seven SI base units.
Its symbol is A.
It is named after Andre-Marie Ampere (1775–1836),
the French physicist and mathematician,
considered as the father of electrodynamics.
André-Marie Ampère
Ampère
The formal definition of ampere
is based on the Ampere’s Circuital Law
which gives the attractive force
between two parallel wires carrying an electric current.
One Ampere is the constant current which will produce
an attractive force of 2 10–7 newton per meter of length
between two straight, parallel, infinitely long conductors
having negligible circular cross section
placed one meter apart in a perfect vacuum.
Ampère
Using the SI definition of electric charge, the Coulomb,
an electric current of one ampere is
one coulomb of charge flowing past a given point /second
Charge Q is determined by steady current I
flowing for a time t as Q = It.
The ampere is a measure of the amount of electric charge
passing a point per unit time.
Ampère
One ampere is equal to around 6.241 1018 (or exactly
6 241 509 479 607 717 888) elementary charge units
such as electrons passing a given point each second.
Other electrical units are all defined in terms of the ampere,
the base unit. For example,
One ampere of current results from a potential distribution of
one volt per ohm of resistance, or from
a power production rate of one watt per volt of potential.
Ampère
In practice, the name Ampere
is often abbreviated as amp.
It is likely that in place of the above idealized definition,
new definition for Ampere may be adopted at the CGPM
(The General Conference on Weights and Measures)
meeting in 2011.
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Ampère`s Circuital Law