Download PPT No. 17 * Biot Savart`s Law- Statement, Proof •Applications of

PPT No. 17
* Biot Savart’s Law- Statement, Proof
•Applications of Biot Savart’s Law
* Magnetic Field Intensity H
* Divergence of B
* Curl of B
Biot Savart’s Law
A straight infinitely long wire
is carrying a steady current I.
Point P is at
a perpendicular distance (AP=) R from the wire.
Consider a small element đℓ
at the point O on the wire.
The line joining points O to P (OP=vector r)
makes an angle θ
with the direction of the current element dℓ .
Biot Savart’s Law
Fig. (a) Magnetic Field dB due to current carrying element
(b) Derivation of dB
Biot Savart’s Law
The magnetic field dB due to the current element of
length dℓ at P is observed to be given by
The product dℓ x r has a magnitude dℓ r sin θ.
It is directed perpendicular to both dℓ and r. i.e.
it is perpendicular to the plane of the paper and going into it,
according to the right handed corkscrew rule
(direction in which a right handed corkscrew advances
when turning from dℓ to r).
Biot Savart’s Law
The expression for the total magnetic field B due to the wire
can be obtained by integrating the above expression as
or equivalently,
It is called as the Biot–Savart law which gives
the Magnetic field B generated by a steady Electric current I
when the current can be approximated as
running through an infinitely-narrow wire.
Biot Savart’s Law
If the current has some thickness i.e. current density is J,
then the statement of the Biot Savart’s law is:
or equivalently,
Where dℓ= differential current length element
dV = volume element
μ0 = the Magnetic constant,
r = displacement vector
= the displacement unit vector,
Biot Savart’s Law
The magnetic field B at a point P due to
an infinite (very long) straight wire
carrying a current I is proportional to I, and
is inversely proportional to
the perpendicular distance R
of the point from the wire.
The vector field B depends on
the magnitude, direction, length, and
proximity of the electric current, and also on
a fundamental constant called the Magnetic constant μ0
Biot Savart’s Law
The Biot–Savart law is fundamental to Magnetostatics
It plays a role similar to Coulomb’s Law in Electrostatics.
The Biot-Savart Law relates Magnetic fields to
the electric currents which are their sources just as
Coulomb’s Law relates electric fields to
the point charges which are their sources.
Biot Savart’s Law
The Biot-Savart Law provides a relation between
the cause (moving charge) and
the effect (magnetic field)
in magnetism.
It is an empirical law
(formulated from the experimental observations)
like the Coulomb’s law.
Both are inverse square laws.
Biot Savart’s Law
In spite of this parallel situation,
one important distinction between
the Coulomb’s law and the Biot Savart’s law is that
the magnetic field B, is in the direction of
the vector cross product dℓ x r i.e.
along the perpendicular direction of the plane constituted by
the current length element dℓ and displacement vector r
while electrostatic field E is along the displacement vector.
Biot Savart’s Law
This necessitates representation of B-Field by
vector notation and
3-D space for its visualization.
The magnetic field B as computed
using the Biot-Savart law
always satisfies Ampere’s Circuital Law and
Gauss Law for Magnetism
Biot Savart’s Law
or
Though the above statement of Biot-Savart law is for
a macroscopic current element,
it can be applied in the calculation of magnetic field
even at the atomic/molecular level
(in which case quantum mechanical calculation or theory
is used for obtaining the current density).
Applications of Biot Savart’s Law
Biot-Savart’s law is stated for
a small current element (Idℓ) of wire –
Not for the extended wire carrying current.
However, magnetic field due to
extended wire carrying current
can be found by using the superposition principle i.e.
the magnetic field is
a vector sum of the fields created by
each infinitesimal section of the wire individually.
Applications of Biot Savart’s Law
For calculating the magnetic field due to
an extended wire carrying current
The point in space at which
the magnetic field is to be computed is chosen,
it is held fixed and integration is carried out
over the path of the current(s)
by applying the equation of Biot Savart’s Law.
Applications of Biot Savart’s Law
Applications of Biot Savart’s Law
Magnetic Field at the Centre of the Current Loop
Consider a circular loop of radius r carrying a current I
At the center of the loop,
the magnitude of the magnetic field B is given by
B=
The direction of the magnetic field is indicated by
The Right Hand Rule
The magnetic field changes away from the center
in both magnitude and direction
Applications of Biot Savart’s Law
Magnetic Field due to a Circular Current Loop
Magnetic field at any point on the axis of a circular loop
can be obtained as follows
Consider a circular loop of radius a having its centre at O.
Point P is situated on the axis of loop at a distance R
from the centre O of the loop.
The loop carries a current I.
Magnetic Field due to a Circular Current Loop
The magnitude of the fields dB & dB’ due to
small current elements dℓ and dℓ’ of the circle,
centered at A and A’
(at diagrammatically opposite points) respectively
is given by Biot Savart’s law as
Magnetic Field due to a Circular Current Loop
Fig. Magnetic B-Field due to a circular current loop
Magnetic Field due to a Circular Current Loop
The direction of the field dB is normal to a plane containing
dℓ and AP i.e. along PQ and that of dB’ is along PQ’.
The fields can be resolved into two components in mutually
perpendicular directions along the axis and
Perpendicular to axis i.e. along PS/ PS’.
Their Components dB cosφ along PS and
dB’cos φ along PS’ are
equal and opposite and get cancelled.
Components along the axis dB sinφ and dB’ sinφ
both have the same direction and are added up.
This applies to all such pairs of elements.
Magnetic Field due to a Circular Current Loop
Thus the resultant field due to the loop
is directed along the axis of the loop and
its magnitude is obtained by integrating the expression
Magnetic Field due to a Circular Current Loop
The magnetic field B due to the circular current loop
of radius a at a point on its axis and a distance R away
is given by integrating the above expression as
(i
is the unit vector along OP, the x-axis)
Some other examples of geometries where
the Biot Savart’s Law can be used to advantage
in calculating the Magnetic field resulting from
an Electric current distribution are as follows
Applications of Biot Savart Law
Magnetic Field of an Infinitely Long Wire
The magnetic field B at a point distance r from
an infinitely long wire carrying current I has magnitude
The direction of Magnetic Field
is given by the Right-hand rule.
Applications of Biot Savart Law
Magnetic Field of a Long Solenoid
The magnetic field B inside the long solenoid of length L
with N turns of wire wrapped evenly along its length is
uniform throughout the volume of the solenoid (except
near the ends where the magnetic field becomes weak)
and is given by
B is independent of the length and diameter and
uniform over cross-section of solenoid
Applications of Biot Savart Law
Magnetic Field of a Solenoid
Magnetic Field inside A long Solenoid
(A) Divergence of B-field
According to the Gauss law in electrostatics,
divergence of the static electric field is equal to
the total density of a stationary electric charge/s
at a given point.
div. E =
(A) Divergence of B-field
However in magnetostatics a magnetic charge
(i.e. monopole) is not found to exist.
(The source of magnetic fields is
moving electric charges, Not the static ones).
Due to the absence of magnetic charges,
the magnetic field is divergenceless.
In Differential form
(where B is the Magnetic field
denotes Divergence)
(A) Divergence of B-field
This is called as the Gauss's law for magnetism
(though this term is not universally adopted).
It states that the magnetic field B
has divergence equal to zero i.e.
magnetic field is a solenoidal vector field
(A) Divergence of B-field
It is equivalent to the statement that
Magnetic Monopole
(i.e. isolated North or South magnetic pole)
does not exist.
The basic quantity for magnetism is
the Magnetic Dipole
Not the magnetic charge or monopole.
Hence, the law is also called as
"Absence of Free Magnetic poles ".
(A) Divergence of B-field
The statement of Gauss's law for magnetism
in integral form is given as
Where S is any closed surface
(the boundary enclosing a three-dimensional volume);
dA is a vector, having magnitude equal to
the infinitesimal area of the surface S and
direction along the surface normal pointing outward.
(A) Divergence of B-field
The left-hand side of the equation in integral form denotes
the net flux of the magnetic field out of the surface.
The law implies that the net magnetic flux
into and out of a volume is zero.
Thus Gauss's law for magnetism can be written in
both- differential and integral- forms.
These forms are equivalent due to the Divergence theorem
(A) Divergence of B-field
The magnetic field B, like any vector field,
can be represented by field lines.
Gauss's law for magnetism also implies that
the field lines have neither a beginning nor an end.
They either form a closed loop,
or extend to infinity in both directions.
(B) Curl of B-field
Circulation is the amount of
pushing, twisting or turning force
along a closed boundary / path
when the path is shrunk down to a single point.
Circulation is the integral of a vector field along a path.
A vector field is usually the source of the circulation.
Curl is the circulation per unit area,
circulation density, or
rate of rotation (amount of twisting at a single point
(B) Curl of B-field
The curl of a force F
is calculated as follows
Let the Force at position r=
Direction at position r =
Total pushing force =
Curl =
(B) Curl of B-field
Curl is defined as the vector field having
magnitude equal to the maximum "circulation" at each point
and to be oriented perpendicularly
to this plane of circulation for each point.
The magnitude of
is the limiting value of circulation per unit area
(B) Curl of B-field
=> the field is said to be an irrotational field.
The physical significance of the curl of a vector field is
the amount of "rotation" or angular momentum of
the contents of given region of space.
It arises in fluid mechanics and elasticity theory.
It is also fundamental in the theory of electromagnetism
(B) Curl of B-field
In magnetostatics, it can be proved that
the curl of magnetic field B is given by
Thus the curl of a magnetic B field at any point is equal to
μ0 times the current density J at that point.
This simple statement relates the magnetic field
and moving charges.
The equation is mathematically equivalent to
the line integral equation given by Ampere’s law.
Divergence and Curl of B-field
The equations in terms of
Divergence and Curl of magnetic B-field
are also called as the laws of Magnetostatics.
They correspond to
the curl and divergence of
electric field E respectively in electrostatics as follows
Divergence and Curl of B-field
Electrostatics
Field is without curl
Magnetostatics
Field is without divergence
Field B–Source j relation
Field E– Source ρ relation
Divergence and Curl of B-field
The equations for divergence and curl for vector fields
are extremely powerful.
Expressions for divergence and curl of a magnetic field
describe uniquely any magnetic field from
the current density j in the field in the same manner that
the equations for the divergence and curl for the electric field
describe an electric field from
the electric charge density ρ in the electric field.
Divergence and Curl of B-field
The four equations involving Curl and Divergence for
Electric and Magnetic fields are
the versions of Maxwell’s equations for
static electromagnetic fields.
They describe mathematically the entire content of
Electrostatics and Magnetostatics.