The Biot–Savart Law
The Biot–Savart Law
• The vector dB is perpendicular both to vector ds (which points in the direction of the
current) and to the unit vector rˆ directed from ds toward P.
2
• The magnitude of dB is inversely proportional to r , where r is the distance from
ds to P.
• The magnitude of dB is proportional to the current and to the magnitude ds of the
length element ds.
• The magnitude of dB is proportional to sin q, where q is the angle between the
Vectors ds and dB .
These observations are summarized in the mathematical expression known today
as the Biot–Savart law:
r
Magnetic Field Surrounding a
Thin, Straight Conductor
Consider a thin, straight wire carrying a constant
current
I and placed along the x axis as shown in the Figure
Determine the magnitude and direction of the
magnetic
field at point P due to this current.
Magnetic Field Surrounding a
Thin, Straight Conductor
The right-hand rule
for determining the direction of the
magnetic field surrounding a long,
straight wire carrying a current.
Note that the magnetic field lines
form circles around the wire.
Magnetic Field Due to a
Curved Wire Segment
Calculate the magnetic field at point o for the current carrying
wire segment shown in the Figure.
The wire consists of two straight portions and
a circular arc of radius R, which subtends
an angleq. The arrowheads on the wire
indicate the direction of the current.
Magnetic Field on the Axis of
a Circular Current Loop
 Consider a circular wire loop of radius R located in the yz
plane and carrying a steady current I, as in the Figure
 Calculate the magnetic field at an axial point P a distance
x from the center of the loop.
Magnetic Field on the Axis of
a Circular Current Loop
Magnetic Field on the Axis of
a Circular Current Loop
Magnetic Field at the center
Axis of a Circular Current
Loop
What if you were asked to find the magnetic field
at the center of a circular wire loop of radius R that
carries a current I?
Can we answer this question at this point in our
understanding of the source of magnetic fields?
Magnetic Field Lines
(a) Surrounding a current loop.
(b) Surrounding a current loop,displayed with iron filings.
(c) Surrounding a bar magnet.
Note the similarity between this line pattern and that of a
current loop
The Magnetic Force Between Two
Parallel Conductors
Two parallel wires that each carry a
steady current exert a magnetic force on
each other.
The field B2 due to the
current in wire 2 exerts a magnetic force
of magnitude F  I1l  B2 on wire
o I 2
F1  I1l
2 a
The force:
attractive if the currents are parallel
repulsive if the currents are ant parallel.
Quick Quiz
 For I1 = 2 A and I2 = 6 A, which is true:
 (a) F1 = 3F2, (b) F1 = F2/3, (c) F1 = F2?
Ampère’s Law
(a) When no current is present in the wire, all
compass needles point in the same direction
(toward the Earth’s north pole).
(b) When the wire carries
a strong current, the compass needles deflect in a
direction tangent to the circle, which is the direction
of the magnetic field created by the current
 The line integral of B  ds
around any closed path
equals o  I , where  I is the total steady current
passing through any surface bounded by the closed
path.
o  I
 B.ds  B  ds  2 r 2 r  o  I
Quick Quiz
 Rank the magnitudes of B( ds for the closed paths in
from least to greatest.
Quick Quiz
Several closed paths near a
single current-carrying wire.
Rank the magnitudes of  B.ds
for the closed paths in from
least to greatest.
Examp.30.4 The Magnetic Field Created
by a Long Current-Carrying Wire
 A long, straight wire of radius R carries a
steady current I that is uniformly
distributed through the cross section of
the wire the Fig. Calculate the magnetic
field a distance r from the
center of the wire in the regions
r  R And r  R .
Solution
The Magnetic Field Created by a Toroid
A device called a toroid (Fig.) is often used to create an
almost uniform magnetic field in some enclosed area.
The device consists of a conducting wire wrapped
around a ring (a torus) made of a nonconducting
material.
Example 30.5
For a toroid having N closely spaced turns of
wire, calculate the magnetic
field in the region occupied by
the torus, a distance r from
the center.
Solution
Example 30.6
Magnetic Field Created by an
Infinite Current Sheet
Solution
Example 30.7
The Magnetic Force on a Current
Segment
Solution
The Magnetic Field of a
Solenoid
Magnetic Flux
Magnetic flux through a plane
lying in a magnetic field
(a)The flux through the plane is zero when the
magnetic field is parallel to the plane surface.
(b) The flux through the plane is a maximum
when the magnetic field is perpendicular to the
plane.
Example 30.8
 Magnetic Flux Through a Rectangular Loop
Gauss’s Law in Magnetism
 the net magnetic flux through any closed surface is
always zero:
The magnetic field lines of a
bar magnet form closed loops
 The net magnetic flux through a closed surface
surrounding one of the poles (or any other closed
surface) is zero.
(The dashed line represents the intersection of the
surface with the page.)
The electric field lines surrounding
an electric dipole
begin on the positive
charge and terminate on
the negative
charge.
The electric flux through a
closed surface urrounding
one of the charges is not
zero.