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Magnetic Field on the Axis of a Current Loop

Submitted by Rich on Thu, 06/15/2017 - 22:17

In this lesson students will find that a current-carrying loop can be regarded as a magnetic dipole, as it generates a magnetic field for points on its axis.  The figure below shows a diagram and the equation for the magnetic field B.  Derivation of this equation requries knowledge of the Biot-Savart Law, calculus and trigonometry.  But in this lesson we are interested only in comparing experimental results from PocketLab's magnetometer to the theoretical equation in the figure below.  More advanced students can consider derivation of the equation, if they wish.

Equation

There are many ways that you can make a current loop.  The author used a plastic ribbon spool approximately 3" in diameter and 3/4" wide, and then wrapped 10 turns of insulated wire around the spool.  The ends of the wire were connected to a DC power supply that supplied constant current for the current loops.  The photo below shows PocketLab with its magnetometer centered in the middle of the spool on the axis of the spool.  PocketLab's magnetic sensor is located about 0.5 cm in from the its edge, shown by the black X drawn on PocketLab.  PocketLab is set to provide magnetic field magnitude data and is zeroed when there is no current in the wire loops.

Current Loops

The two photos below provide two more views of the apparatus setup.  It is important to keep the magnetic sensor on the loop axis, as it is moved to known distances from the center of the loop.  The author used some small blocks of wood for this purpose.  A meter stick with its zero point at the center of the loops allows moving PocketLab gradually outward, increasing the value of x by one cm for each move of PocketLab.  The author's setup used a current of i = 5.12 amp, R = 0.0361 m, with x varying from 0.00 m to 0.10 meters by steps of 0.01 m each.  The number of loops N = 10.

Current Loops

Current LoopsThe graph shown below, constructed in Excel, was obtained from data in the magnetometer.csv file produced by the PocketLab app.  The horizontal plateaus are labeled with the average value of the magnetic field for each plateau.  These averages could be obtained using Excel, but are much easier and quicker to obtain using Logger Pro, an exceptional educational data analysis software package produced by Vernier Software & Technology (vernier.com).  The highest plateau is where PocketLab was at x=0.  The next plateau is for x=0.01 m, the next for x=0.02 m, and so on, through x=0.10 m.

Magnetometer Magnitude vs Time

The graph below, produced in Excel, summarizes the experimental results for magnetic field vs. distance along the axis, as they compare to the expected results from the theoretical equation.  A good discussion would be for students to suggest reasons for discrepancies between theory and experiment.

Magnetic Field vs X

Magnetometer Magnitude vs Time
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