Helmholtz coil
As a Helmholtz coil is called a special coil assembly on the German physicist Hermann von Helmholtz (1821-1894) back: Two short coil with a large radius R are placed parallel at a distance R on the same axis and the same sense current flowing through it ( at gegensinnigem current flow see Maxwell coil).
The field of each coil is inhomogeneous. The superposition of both fields results between two coils close to the coil axis, an area with a largely homogeneous magnetic field, which is freely available for experimentation.
There are Helmholtz coils in different designs: cylindrical, square, as well as three orthogonal established couples ( three-dimensional). With the three-dimensional array can be generated between the coil pairs a magnetic field of arbitrary direction and thus examining an item without having to rotate it by varying the current ratio.
Applications of the Helmholtz coil
- Determination of the spec. Electron charge, according to Helmholtz with fine beam
- Quality control of permanent magnet
- Hall - effect studies
- Production of field-free spaces through targeted screening of the Earth's magnetic field
- Calibration of magnetic probes and magnetometers
- RF coil for magnetic resonance imaging
- Gradient coil ( as Maxwell coil) in magnetic resonance imaging
- Magnetic Therapy
Benefits
The magnetic field strength is generated - as with any air coil - not only linearly dependent on the coil current, but less location-dependent than in a single narrow coil. From the coil geometry, the current and the number of turns, the magnetic field strength along the axis can be calculated analytically. Therefore, the Helmholtz coil is ideal for the calibration of magnetometers, such as fluxgate magnetometers used.
Calculating the magnetic flux density
If the origin of the coordinate system is located in the center of the coil, resulting with the Biot- Savart law for the magnetic flux density in vacuum along the symmetry axis for the special case of only one turn (N = 1):
The flux density in the center of the Helmholtz coil pair is the superposition of two circular currents in the ratio:
Where μ0 the magnetic permeability of free space, I is the coil current, and R is the coil radius.
It can also determine the overall flux density of Helmholtz coils. The Helmholtz coils are composed of two conductor loops ( Strömfäden ) with the number of turns N1 and N2, by which a respective current I1 and I2 flows. There is therefore a current density ( in cylindrical coordinates ) by:
Here, the center of each coil is located in or on the x-axis.
Using the Biot- Savart law can be the vector potential of the coil calculate:
In this case, ie the point at which the vector field is determined, the integration and the field point. For the vector potential thus results while performing the trivial integrations:
With the magnetic flux density can be calculated. For this, the vector potential will be decomposed into the components of the Zylinderszmmetrie:
So that is valid for the individual components
With the rotation operator in cylindrical coordinates
Can now calculate the flux density:
Accordingly, the Cartesian components result to
The thus found analytical expressions can be only in certain cases further simplified, since the elliptic integrals contained can only be calculated numerically.
In the event that only the coil axis is considered ( ) results and
In coils origin () shall apply mutatis mutandis and
Pair of Helmholtz coils, each with N turns
Example: I = 1.7 A; N = 130; R = 0.15 m
Field profiles at various coil separations
In the arrangement of the Helmholtz disappears in the middle of the first and second derivative of the box function according to the sides of the field strength drops off relatively quickly. This can be seen in the gallery below.
Larger distances result in a larger experimental volume, but at the coil center sloping down towards the field strength values . Smaller intervals yield higher field strengths, but a smaller experimental volume and the measured values are in good agreement with the calculated values . With iron filings can be shown in the vicinity of the coil axis in the experiment, the good homogeneity of the magnetic field.
Maxwell coil
If the current flows through the coils in opposite directions, so at suitably chosen geometry inside a constant field gradient is generated. A is the distance of the coils to each other, then disappears at x = 0, the second and the third derivative, the function field is thus there is a straight line. Is called the coil assembly then Maxwell coil, sometimes anti-Helmholtz coil.
The calculation of the field profile along the axis of symmetry ( x-axis) is at an entirely analogous manner as in the case of the same direction of the circulating currents. Obtained for pairs of coils with the same number of turns N:
Then for the field gradient in the center:
Photo Gallery
The following measured or calculated field profiles are presented for Helmholtz coils:
Field profile in the direction of the coil axis (measured )
Field profile transverse to the direction of the coil axis (measured )
Magnetic field of the Helmholtz coil made visible with iron filings
Calculated magnetic field of the Helmholtz coil
Amount of the magnetic flux density as a function of position. The section plane passing through the center.
Maxwell coil: field strength with optimized coil spacing of and individual fields in arbitrary units.