<< Chapter < Page Chapter >> Page >

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 is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Electricity and magnetism. OpenStax CNX. Oct 20, 2009 Download for free at http://cnx.org/content/col10909/1.13
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Electricity and magnetism' conversation and receive update notifications?