Now that you have become familiar with the BiotSavart Law for calculating the magnetic field around a currentcarrying 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 BiotSavart 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.
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 BiotSavart 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.
