Details of the Topic :
A) Oerested’s Experiment
B) Biot - Savart Law
C) Applications of Biot - Savart Law
D) Magnetic Dipole moment
E) Ampere’s Circuital law & its applications
F) Force between two parallel current carrying wire
A) Oersted's Experiment
While performing his electric demonstration, Oersted noted to his surprise that every time the electric
current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but
in the months that followed worked hard trying to make sense out of the new phenomenon.
What Oersted saw...
But he couldn't! The
wire nor repelled from
angles (see drawing
findings (in Latin!)
needle was neither attracted to the
it. Instead, it tended to stand at right
below). In the end he published his
without any explanation.
B) Biot-
Savart Law
The magnetic equivalent of Coulomb's law is the Biot-Savart law for the magnetic field
produced by a short segment of wire, , carrying current I:
where the direction of is in the direction of the current and where the vector points from the
short segment of current to the observation point where we are to compute the magnetic field.
Since current must flow in a circuit, integration is always required to find the total magnetic field
at any point. The constant is chosen so that when the current is in amps and the distances are
in meters, the magnetic field is correctly given in units of tesla. Its value in our SI units is
exactly
A quick comparison of this value with the Biot-Savart law probably makes you wonder what role
is supposed to play here. It plays the same role it did in Coulomb's law: it was required in
Coulomb's law so that Gauss's law wouldn't have a
, and it is required in the Biot-Savart law
so that Ampere's law won't have one either.
There are two simple cases where the magnetic field integrations are easy to carry out, and
fortunately they are in geometries that are of practical use. We use the formula for the magnetic
field of an infinitely long wire whenever we want to estimate the field near a segment of wire,
and we use the formula for the magnetic field at the center of a circular loop of wire whenever
we want to estimate the magnetic field near the center of any loop of wire.
Similarities and Differences Between Biot-Savart's
Law and Coulomb's Law
Similarities
1) Both magnetic and electric field depend inversely on the distance between the source and the field point
2) Both are long-range forces
3) Principle of superposition applies to both fields as the fields are linearly related to the sources.
Differences
C)
APPLICATIONS of BIOT SAVART LAW
a) Infinitely Long Wire: The magnetic field at a point a distance r from an infinitely long wire
carrying current I has magnitude
and its direction is given by a right-hand rule: point the thumb of your right hand in the
direction of the current, and your fingers indicate the direction of the circular magnetic field lines
around the wire.
b) Circular Loop: The magnetic field at the center of a circular loop of current-carrying
wire of radius R has magnitude
and its direction is given by another right-hand rule: curl the fingers of your right hand in the
direction of the current flow, and your thumb points in the direction of the magnetic field inside
the loop.
c) Long Thick Wire: Imagine a very long wire of radius a carrying current I distributed
symmetrically so that the current density, J, is only a function of distance r from the center of the
wire. Ampere's law can be used to find the magnetic field at any radius r. Outside the wire,
where
, we have
just as if all the current were concentrated at the center of the wire. Inside the wire, where r < a,
we have
where I(r) is the current flowing through the disk of radius r inside the wire; the current outside
this disk contributes nothing to the magnetic field at r. Note that this is analogous to the result for
symmetric electric fields, discussed in Chapter 24.
d) Long Solenoid: Imagine a long solenoid of length L with N turns of wire wrapped evenly
along its length. Ampere's law can be used to show that the magnetic field inside the solenoid is
uniform throughout the volume of the solenoid (except near the ends where the magnetic field
becomes weak) and is given by
where n = N/L.
e) Toroid: Imagine a toroid consisting of N evenly spaced turns of wire carrying current I.
(Imagine winding wire onto a bagel, with the wire coming up through the hole, out around the
outside, then up through the hole again, etc..) Ampere's law can be used to show that the
magnetic field within the volume enclosed by the toroid is given by
where R is the distance from the z-axis in cylindrical coordinates, with the z-axis pointing
D)
Magnetic Dipole Moment
For the current loop, the magnetic dipole moment would point through the loop (according to the right
hand grip rule), with a magnitude equal to the current in the loop times the area of the loop.
In the simplest case of a planar loop of electric current, its magnetic moment is defined as:
M = I *A
where
M is the magnetic moment, a vector measured in ampere–square meter, or equivalently in joule
per tesla,
A is the vector area of the current loop, measured in square meter (x, y, and z coordinates of this
vector are the areas of projections of the loop onto the yz, zx, and xy planes), and
I is the current in the loop (assumed to be constant), a scalar measured in ampere.
By convention, the direction of the vector area is given by the right hand grip rule (curling the fingers of
one's right hand in the direction of the current around the loop, when the palm of the hand is "touching"
the loop's outer edge, and the straight thumb indicates the direction of the vector area and thus of the
magnetic moment).
Ampere’s Circuital law
Ampere's law is a useful relation that is analogous to Gauss's law. Ampere's law is a relationship
between the tangential component of magnetic field at points on a closed curve and the net
current through the area bounded by the curve.
Ampere's law is formulated in terms of the line integral of B around a closed path denoted by
We divide the path into infinitesimal segments dl and for each one calculate the scalar product of
B and dl. In general, B varies from point to point and the B at the location of each dl must be
used.
Consider a long straight conductor carrying a current passing through the centre of a circle of
radius r in a plane perpendicular to the conductor.
Using Biot Savart's law we know already that the field at a distance r is
the field at all points on the circle and the direction is given by the tangent drawn to the circle at
that point.
Magnetic Field of a Solenoid
A solenoid is a tightly wound helical coil of wire whose diameter is small compared
to its length. The magnetic field generated in the centre, or core, of a current carrying
solenoid is essentially uniform, and is directed along the axis of the solenoid. Outside
the solenoid, the magnetic field is far weaker. Figure 27 shows (rather schematically)
the magnetic field generated by a typical solenoid. The solenoid is wound from a
single helical wire which carries a current . The winding is sufficiently tight that
each turn of the solenoid is well approximated as a circular wire loop, lying in the
plane perpendicular to the axis of the solenoid, which carries a current . Suppose
that there are such turns per unit axial length of the solenoid. What is the magnitude
of the magnetic field in the core of the solenoid?
Figure 27: A solenoid.
In order to answer this question, let us apply Ampère's circuital law to the rectangular
loop
. We must first find the line integral of the magnetic field around
.
Along
and
the magnetic field is essentially perpendicular to the loop, so there is
no contribution to the line integral from these sections of the loop. Along
the
magnetic field is approximately uniform, of magnitude , say, and is directed
parallel to the loop. Thus, the contribution to the line integral from this section of the
loop is
, where is the length of
. Along
the magnetic field-strength is
essentially negligible, so this section of the loop makes no contribution to the line
integral. It follows that the line integral of the magnetic field around
is simply
(178)
By Ampère's circuital law, this line integral is equal to
times the algebraic sum of
the currents which flow through the loop
. Since the length of the loop along the
axis of the solenoid is , the loop intersects
turns of the solenoid, each carrying
a current . Thus, the total current which flows through the loop is
. This
current counts as a positive current since if we look against the direction of the
currents flowing in each turn (i.e., into the page in the figure), the loop
circulates
these currents in an anti-clockwise direction. Ampère's circuital law yields
(179)
which reduces to
(180)
Thus, the magnetic field in the core of a solenoid is directly proportional to the
product of the current flowing around the solenoid and the number of turns per unit
length of the solenoid. This, result is exact in the limit in which the length of the
solenoid is very much greater than its diameter.
Magnetic Field due to Toroidal Solenoids
Toroid is a hollow circular ring (like a medu vadai) on which a large number of turns of a wire are wound.
The above figure represents a toroid wound with a wire carrying a current I. Consider path 1, by symmetry , if there is
any field at all in this region, it will be tangent to the path at all point and
will equal the product will equal the
product of B and the circumference d = 2pr of the path. The current through the path however is zero and hence from
Ampere's law the field B must be zero.
Similarly, if there is any field at path 3, it will also be tangent to the path at all points. Each turn of the winding passes
twice through the area bounded by this path, carrying equal currents in opposite directions. The net current though
the area is therefore zero and hence B = 0 at all points of the path.
The field of the toroidal solenoid is therefore confined wholly to the space enclosed by the windings.
If we consider path 2, a circle of radius r, again by symmetry the field is tangent to the path and
Each turn of the winding passes once through the area bounded by path 2 and total current through the area is NI,
where N is the total number of turns in the windings.
Using Ampere's law
If the radial thickness of the core is small, field is almost constant across the section.
Here 2pr circumferential length of to the toroid.
Conclusion:
Field outside the toroid and inside the core of the toroid is zero and within the toroid = m0ni
Force between two parallel wires
An interesting effect occurs if we consider two long straight parallel wires separated by a distance d
carrying currents I1 and I2 . Let us examine the case where the two currents are in the same direction, as
in Fig. 1.9.
Figure 1.9: Force between two long straight parallel wires
Wire #2 will experience a magnetic field of Eq.(1.6) due to wire #1 given by
(10)
B1 =
in the direction indicated, and hence will experience a force per unit length given by Eq.(1.4):
(11)
=
The direction of
.
indicated shows that wire #2 will be attracted towards wire #1. In a similar manner,
one can show that wire #1 will experience a force due to the magnetic field
of wire #2, and that
this force
will have a magnitude equal to that of F2 given in Eq.(1.11) but opposite in direction.
Thus, wire #1 will be attracted towards wire #2.
It is a good exercise to show that if the wires were carrying currents in the opposite directions
that the resulting forces will have the same magnitude as in Eq.(1.11) but are such as to cause a
repulsion between the wires.
This force between two current carrying wires gives rise to the fundamental definition of the
Ampère:
If two long parallel wires 1 m apart each carry a current of 1 A, then the force per unit length on each
wire is 2 x 10- 7 N/m.
This definition of the Ampère then gives rise to the basic definition of the unit of charge, the Coulomb:
A wire carrying a current of 1 A transports past a given point 1 C of charge per second.
This definition also explains the reason why the constant
m/A.
of Eq.(1.7) was given exactly as 4
x 10- 7 T