9867

Induction of a Sinusoidal Magnetic Field within a Coil

When a coil is placed in a sinusoidally varying magnetic field, a current and voltage are induced in accordance with Faraday's law. This Demonstration shows plots of the magnetic field. Assuming a uniform near field, the magnetic field is assumed to be uniform everywhere in the space. The coil has a substantial inductance with current or voltage, which can be detected by an oscilloscope. The current and voltage waveforms can be calculated for four types of detection circuits: open circuit, usual oscilloscope impedance (1 M//100 pF), cable's matching impedance (50), and shorted condition. The amplitude of the exciting magnetic field is fixed at 1 A/m, but the frequency and the spacial direction are variable. You can vary the coil configuration (radius, winding cross-sectional diameter, and number of turns). The sideways view of the magnetic field pattern modified by the coil current at the selected time is displayed by using color and arrows.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: a five-turn coil is placed upright to the field of 0.3 MHz and is shorted
Snapshot 2: a five-turn coil is connected to 1M–100pF detection circuit for the field of 0.3 MHz
Snapshot 3: an increased-turn (30) coil resonates with the capacitor of 1M–100pF detection circuit for the 0.3MHz field
The sinusoidal field induces the electromotive force (EMF) in a circular coil of radius with turns, being the angle between field and coil axis. Assuming the circular section of coil has diameter , the coil's inductance is approximated by . Then the current and voltage in the detection circuit are calculated by , , where is the impedance of the detection circuit. For the open circuit, is set at 1 M and 100 pF; for the shorted circuit, it is 50.
The coil current produces a magnetic field superimposed on the original field. The additional magnetic field for the ring current involves Bessel functions. The magnetic field pattern is readily obtained, specifying the time and phase values.
Among the four detection circuits, the detected current increases most generally in this order: open circuit, 1 M//100 pF, 50, and shorted conditions. The detected voltage tends to vary in the opposite direction. However, in the detection circuit of 1 M//100 pF, an extraordinary large current and voltage can occur during resonance between the coil inductance and the capacitor in the detection circuit, as shown in snapshot 3.
Reference
[1] J. D. Jackson, Classical Electrodynamics, 3rd ed., New York: John Wiley & Sons, 1998.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+