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We have already determined magnetic field due to current in circular wire at its center. The approach to determine magnetic field at an axial point is similar. We begin with magnetic field due to small current element and then try to integrate the Biot-Savart expression for the small magnetic field for the entire circle following superposition principle.

This extension of earlier procedure, however, demands a bit of extra three dimensional imagination to arrive at the correct result. In this module, we shall attempt to grasp three dimensional elements as clearly as possible with figures. Let us have a look at the differential Biot-Savart expression :

đ B = μ 0 4 π I đ l X r r 3

There are three vector quantities d B , d l and r . We investigate the spatial relation among these quantities for magnetic field at an axial point.

Magnetic field on an axial point

The magnitude of magnetic field due to current in a current element is given by :

đ B = μ 0 4 π I đ l sin θ r 2

In order to evaluate magnetic field due to complete circular wire, we need to set up corresponding integral properly with respect to various elements constituting the expression. In following subsections, we study these elements in which point of observation is a point on axial line.

The angle between current length element and displacement vectors

The angle (θ ) as appearing in the Biot-Savart expression between current length element vector d l and displacement vector r is right angle. See figure. This right angle should be distinguished with acute angle φ, which is the angle between OA and AP as shown in the figure.

The angle between current length element and displacement vectors

The angle between current length element and displacement vectors is right angle.

The above fact reduces Biot-Savart expression to :

đ B = μ 0 4 π I đ l sin 90 r 2 = μ 0 4 π I đ l r 2

This simplification due to enclosed angle being right angle is true for all points on the circle.

Magnitude of magnetic field

All current elements are at equal linear distance from point P. As a result, the magnitude of magnetic field at P due to any of the equal current elements is same.

đ B 1 = đ B 2 = .

Direction of elemental magnetic field

Unlike enclosed angle (θ), linear distance (r) and magnitude of magnetic field, the direction of magnetic field due to current elements are not same. As such, we can not integrate Biot-Savart differential expression to determine net magnetic field at P. Let us investigate the direction of magnetic fields due to two diametrically opposite current elements. Let the circular wire lie in yz plane as shown in the figure.

Direction of elemental magnetic field

Magnetic field is perpendicular to plane formed by current length element and displacement vectors.

The current length vector d l 1 and displacement vector r 1 form a plane shown as plane 1 and the magnetic field due to current element, B 1 , is perpendicular to plane 1. Similarly, the current length vector d l 2 and displacement vector r 2 form a plane shown as plane 2 and the magnetic field due to current element, B 2 ), is perpendicular to plane 2. Clearly, these magnetic fields are directed in three dimensional space. If we imagine magnetic fields due to other current elements of the circular wire, then it is not difficult to imagine that these elemental magnetic fields are aligned on the outer surface of a conic section and that they are not in same plane.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
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Akash Reply
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s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Damian Reply
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abeetha Reply
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Smarajit Reply
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Source:  OpenStax, Electricity and magnetism. OpenStax CNX. Oct 20, 2009 Download for free at http://cnx.org/content/col10909/1.13
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