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

 

 

 

 

 

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.