Resource Lesson
Magnetic Field Along the Axis of a Current Loop
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Now that you have become familiar with the Biot-Savart Law for calculating the magnetic field around a current-carrying wire and
at the center of a current loop
, let's expand our investigations to calculations of the magnetic field along the axis of a current loop.
In the
following shockwave animation
, a continuous current in a horizontal loop has be "divided" into multiple "current elements." Using the principle of superposition and the Biot-Savart Law each discrete element generates its own magnetic field which, when integrated, produce a resultant field that is aligned parallel to the axis of the loop.
images courtesy of
MIT open courseware
Derivation of the Magnetic Field Along the Axis of a Current Loop
Note in the diagram that the magnetic field contribution,
dB
, of each current segment,
, is perpendicular to the radius vector
.
Let's begin with a basic statement of the Biot-Savart Law.
As shown in the animation, the components perpendicular to the loop's axis,
dB
_{y}
, will cancel as we integrate around the loop. Thus, we will focus on only the horizontal components,
dB
_{x}
.
Using the Pythagorean Theorem, we can express
r
in terms of
x
and
R
,
giving us
Our last step is to calculate the resultant magnetic field by adding up all of these contributions.
Notice that when x = 0, this formula reduces to our former expression for the magnetic field at the center of a current loop derived in an
earlier lesson
.
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