helmholtz_coil

# Helmholtz coil

 Stan Zurek, Helmholtz coil, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/helmholtz_coil

Helmholtz coil, Helmholtz coils, or Helmholtz pair - a pair of coils used typically as a precise source of magnetic field when driven from a precise source of electric current, because the value of magnetic field strength H or magnetic flux density B at the centre of the pair can be calculated from analytical equations.1)2)3)4)

Helmholtz coils are used for local cancellation of Earth's magnetic field (or any other external field), as well as measurement of magnetic dipole moment.5)

Helmholtz coil was named after the German physicist Hermann von Helmholtz (1821-1894).

Helmholtz coil is a precise source of magnetic field at its geometrical centre (black dot); in this drawing N denotes the number of turns in each half of the coil A Helmholtz coil comprises two identical coils resembling circular current loops, positioned parallel to each other on the same axis, and separated precisely by the radius of the circle. The implicit assumption is that each sub-coil or half-coil is infinitely “thin” and “narrow”, so that the cross-section of the bundle of wires is much smaller that the diameter of the coil. The equations for more realistic coils with rectangular cross-section are given in the next section.

For two “thin” half-coils, each with radius r and each comprising number of turns Neach, when connected in series, the value of magnetic field at the geometrical centre can be calculated from the following formula:6)

(1) Magnetic field at the centre of a Helmholtz coil
(turns per half-coil) $$H_{centre}(t) = \frac{N_{each}·I(t)·8}{r·\sqrt{125}} \approx 0.71554·\frac{N_{each}·I(t)}{r}$$ (A/m)
(total turns in both coil halves) $$H_{centre}(t) = \frac{N_{total}·I(t)·4}{r·\sqrt{125}} \approx 0.35777·\frac{N_{total}·I(t)}{r}$$ (A/m)
where: $N_{each}$ - number of turns of each half-coil (unitless), $N_{total}$ - total number of turns of both half-coils (unitless), such that $N_{total}=2·N_{each}$, $I(t)$ - current (A) as a function of time $t$ (s), $r$ - radius of each half-coil and spacing between them (m)
Note: these equations are valid for the half-coils connected in series. For parallel connection the current in each half-coil should be used, rather than the total supplied current.

The relationship is direct, so the instantaneous values of magnetic field follow the instantaneous values of current. This is applicable only for frequencies for which the transmission line effects can be ignored (typically wavelength of frequency should be at least 10 times greater than the dimensions of the coil).

### Calculator of H at the centre of a Helmholtz coil This section is an interactive calculator.
Magnetic field strength H at the centre of a Helmholtz coil
$$H_{centre}(t) = \frac{N_{each}·I (t)· 8 } {r·\sqrt{125}} \approx 0.71554·\frac{N_{each}·I}{r}$$ (A/m)
where: $I(t)$ - current (A) at time $t$ (s), $N_{each}$ - number of turns (unitless) of each half-coil, $r$ - radius (m)
 Current I = (nA) (μA) (mA) (A)      Radius r = (mm) (cm) (inch) (m) Choose option: turns per half-coil turns total      Number of turns N = (unitless) H = (A/m) (kA/m) (A/cm)        B (μ0) = (T) (mT) (uT) Notes: This equation is valid only for infinitely “thin” coils. The instantaneous values of H are directly proportional to the instantaneous values of I. The value of B(μ0) is for vacuum (i.e. non-magnetic medium).

## Field along axis

Field along the axis of a round Helmholtz coil, shown as a sum of the field produced by each half-coil (assumed to be a “thin” current loop); the horizontal values are normalised so the diameter d=1 (and radius r=0.5), and the vertical values to the field at the centre of the Helmholtz pair The analytical equation for magnetic field along the axis of a Helmholtz coil can be derived by combining two current loops offset from each other by their radius $r$.7) It can be assumed that the centre is at zero position, the first current loop is offset in the positive direction by half of the required distance (+r/2), and the second half by the same amount in the negative direction (-r/2). The contribution from both coils can be added up, taking into account the number of turns in each half-coil. Therefore, using the equation of a single “infinitely thin” (filamentary) current loop the equation for a “thin” round Helmholtz coil can be derived as equation (2).

Equation (2) simplifies to equation (1) at x = 0.

(2) Magnetic field of an ideal thin round Helmholtz coil along its axis
$$H(t,x) = \frac{I(t)·N_{each}·r^2}{2}· \left[ \left( \left( x + \frac{r}{2} \right)^2 + r^2 \right)^{-3/2} + \left( \left( x - \frac{r}{2} \right)^2 + r^2 \right)^{-3/2} \right]$$ (A/m)
where: $H(t,x)$ - instantaneous values of magnetic field strength H (A/m) in time t (s) and position x (m), $I(t)$ - instantaneous values of electric current I (A), $N_{total}$ - total number of turns (unitless) such that $N_{total} = 2·N_{each}$, $r$ - radius of the Helmholtz coil (m), $x$ - distance from centre along axis (m)
Note: this equation is valid only for filamentary coils, namely each half-coil is an infinitely thin wire.

### Thick coils

Real coils have some finite width w, and some finite thickness resulting from the difference between the outer diameter D and the inner diameter d. If the current distribution is assumed uniform throughout the cross-section of the coil, then the equivalent equation for the field along the axis of a “thick” Helmholtz coil can be derived by combining two "thick" solenoids, with a method analogous to that described above, resulting with equation (3). This was derived by assuming that the inner diameter d is equal to that the of the ideal filamentary coil, which leads to a slightly lower field value at the centre of the coil.

In such an approach, the spacing between the half-coils is defined as the spacing between their symmetry line (and not between their edges).

For even more realistic calculations the current and positioning of each wire has to be taken into account. And because of quantisation errors the finite-element modelling does not have sufficient precision to be used for calculations of such standard coils.8) For precise calculations even the thickness of the insulation and positions of strands should be taken into account.9)

Normalised graphs of magnetic field generated by various “thick” Helmholtz coils; larger ratio of diameters D/d has a larger effect on error than the increased width w (with w = 0.25 the configuration resembles continuous solenoid) (3) Magnetic field strength along the axis of a “thick” round Helmholtz coil
$$H_{thick}(t,x) = \frac{I(t)·N_{each}}{w·(D-d)}· \bigg[ Q_{pp} · \text{ln} (K_{pp}) + Q_{np} · \text{ln} (K_{np}) + Q_{pn} · \text{ln} (K_{pn}) + Q_{nn} · \text{ln} (K_{nn}) \bigg]$$
where: $H_{thick}(t,x)$ - instantaneous values of magnetic field strength (A/m) at the time $t$ (s) and position $x$ (m), $I(t)$ - instantaneous values of current (A) at time $t$ (s), $N_{each}$ - number of turns in each half of the solenoidal coil (unitless), $w$ - width of each half-coil (m), $d$ - inner diameter of the coils (m), $D$ - outer diameters of the coils (m), $x$ - position (m) along the axis such that x = 0 is at the centre between the pair, “ln” - natural logarithm function, “n” - “negative” coefficient, “p” - “positive” coefficient, and:
$Q_{pp} = \frac{w}{2} + \left( x + \frac{d}{4} \right)$ $Q_{np} = \frac{w}{2} - \left( x + \frac{d}{4} \right)$ $Q_{pn} = \frac{w}{2} + \left( x - \frac{d}{4} \right)$ $Q_{nn} = \frac{w}{2} - \left( x - \frac{d}{4} \right)$
$K_{pp} = \frac{D + \sqrt{D^2 + 4·Q_{pp}^2 }} {d + \sqrt{d^2 + 4·Q_{pp}^2 }}$ $K_{np} = \frac{D + \sqrt{D^2 + 4·Q_{np}^2 }} {d + \sqrt{d^2 + 4·Q_{np}^2 }}$ $K_{pn} = \frac{D + \sqrt{D^2 + 4·Q_{pn}^2 }} {d + \sqrt{d^2 + 4·Q_{pn}^2 }}$ $K_{nn} = \frac{D + \sqrt{D^2 + 4·Q_{nn}^2 }} {d + \sqrt{d^2 + 4·Q_{nn}^2 }}$

### Square Helmholtz coil

Schematic drawing of a “thin” square Helmholtz coil; also for a “narrow” coil $w≈0$ The equation for square Helmholtz coil can be derived in a similar way as described above, namely by employing the basis of square current loops or square solenoids.

Set of three large orthogonal Helmholtz coils used for compensation of external magnetic field in 3D10) by A. Jodko-Władzińska, K. Wildner, T. Pałko, M. Władziński, MDPI, CC-BY-4.0

Square Helmholtz coils are used because they have larger volume accessible inside of the coil,11) but at the expense of magnitude lower than for a comparable round coil (by around 5-10%, depending on separation between the half-coils).

The uniformity of the field at the centre is comparable for square and round coils, and can be even slightly better for the square coils.12) Square coils are typically used for cancellation of magnetic field external to the given experiment, when large volume is required. For sufficiently large coils human subject can comfortably sit inside, with an arrangement that the geometrical centre of the coil coincides with the patient's head (as shown in the photograph).

Extremely large coils were constructed to accommodate sea ships for the purpose of demagnetising the hull (also called deperming or degaussing).13)

The parametric equation (4) can be derived for an ideal “thin” and “narrow” square Helmholtz coil.14)

Uniformity requires that the higher order terms in the equation are suppressed, which leads to the requirement that the second derivative is equal to zero,15) and thus the separation coefficient becomes approximately $k ≈ 0.544506$ (see equation (5)), such that coil separation $q = k·a$ (where $a$ - length of the side of the square). Therefore, the square coils are separated by around 10% than equivalent round coils, and thus the field at the centre is lower by a similar amount (with the same number of turns and current).

(4) Magnetic field strength along the axis of a “thin” square Helmholtz coil
$$H_{square,ideal}(t,x) = \frac{I(t)·N_{each}·a^2}{2·π} · \left( \frac{1}{(\frac{a^2}{4} + (x+\frac{q}{2})^2)·\sqrt{\frac{a^2}{2}+(x+\frac{q}{2})^2}} + \frac{1}{(\frac{a^2}{4} + (x-\frac{q}{2})^2)·\sqrt{\frac{a^2}{2}+(x-\frac{q}{2})^2}} \right)$$ (A/m)
where: $H_{square,ideal}(t,x)$ - instantaneous values of magnetic field strength (A/m) at time $t$ (s) and position $x$ (m), $I(t)$ - instantaneous values of current (A), $N_{each}$ - number of turns in each half of the coil (unitless) so that $N_{total} = 2·N_{each}$, $a$ - length (m) of the side of the square, $x$ - position along the axis of the coil (m) such that $x=0$ is at the centre, $q$ - distance between the half-coils (m)
Note - this equation is only valid for ideal “thin” and “narrow” square coils.
(5) Optimum separation coefficient k for a “thin” square Helmholtz coil
$$k = \frac{a}{q} = \sqrt{ \left( \sqrt{ \frac{305}{5832} } + \frac{1}{3} \right)^{1/3} + \frac{7}{18· \left( \sqrt{\frac{305}{5832}} + \frac{1}{3} \right)^{1/3}} -1} ≈ 0.544506$$ (unitless)

#### Alternative equation for square coils

Normalised graphs of magnetic field generated by a square “thin” Helmholtz coil as compared to the ideal “thin” round coil, such that the side $a$ of the square is equal to the diameter $d$ of a round coil; the width $w$ has a relatively small effect on the amplitude, with w=0.25·d denoting continuous solenoid (note the inferior uniformity around the centre due to separation q=a/2) An alternative equation can be derived by using the basis of the “thin” square solenoid, with a finite width w.

In the following equation (6), the separation distance q between the centres of the half coils was assumed to be equal to q which is the half of the side of the square, so that q = a/2.

As described above, for a square coil better field uniformity is achieved for slightly larger separation coefficient (see also section Field uniformity below).

A similar derivation can be made for a different value of separation. The calculated values are then the same, even though the analytical equations differ.

(6) Magnetic field strength along the axis of a “thin” square Helmholtz coil
$$H_{square}(t,x) = \frac{I(t)·N_{each}}{w·π}· \bigg[ \text{atan}(Q_{pp}·K_{pp}) - \text{atan}(Q_{pn}·K_{pn}) + \text{atan}(Q_{np}·K_{np}) - \text{atan}(Q_{nn}·K_{nn}) \bigg]$$
where: $H_{square}(t,x)$ - instantaneous values of magnetic field strength (A/m) at time $t$ (s) and position $x$ (m), $I(t)$ - instantaneous values of current (A), $N_{each}$ - number of turns in each half of the coil (unitless), $w$ - width of each coil (m), $a$ - length (m) of the side of the square (equivalent of diameter $d$), $x$ - position along the axis of the coil (m) such that x = 0 is at the centre between the pair, “atan” - trigonometric function arctangent, “n” - “negative” coefficient, “p” - “positive” coefficient, and:
$Q_{pp} = 2·x+\frac{a}{2}+w$ $Q_{pn} = 2·x+\frac{a}{2}-w$ $Q_{np} = 2·x-\frac{a}{2}+w$ $Q_{nn} = 2·x-\frac{a}{2}-w$
$K_{pp} = \frac{\sqrt{2·a^2 + Q_{pp}^2}}{a^2}$ $K_{pn} = \frac{\sqrt{2·a^2 + Q_{pn}^2}}{a^2}$ $K_{np} = \frac{\sqrt{2·a^2 + Q_{np}^2}}{a^2}$ $K_{nn} = \frac{\sqrt{2·a^2 + Q_{nn}^2}}{a^2}$
Note: in this equation the assumed separation between the coils is too small to achieve optimum field uniformity. See also eq. (4).

### Maxwell coil

Further improvement of uniformity of field over larger volume can be achieved by adding more loops of current, in an appropriate arrangement, distributed over a spherical shape. The Helmholtz pair is effectively also distributed on a sphere, but with the minimum number of loops (just two). At the other extreme is a full sphere, but it is not used in practice due to complexity of making of such coil and the lack of access to its inside.16)

Three such co-axial coils are known as a Maxwell coil. They have to be arranged such that the largest coil in the middle has radius R and the two smaller ones have radius $r = R · \sqrt{4/7}$ and the distance between the larger and the smaller coils should be $q = R · \sqrt{3/7}$. The current can be equal in each coil, but the ratio of ampere-turns must be matched so that outer/inner = 49/64 (if the current is the same then the turns must have such ratio of 49/64).17)18)19)

The name gradient Maxwell coil is also used when referring to an anti-Helmholtz pair separated by the distance of √3·r (see detailed description in the following sections). Such name of Maxwell coil (denoting the gradient coil) is used especially in the particle accelerators,20) and magnetic resonance imaging.21)

### Four and more coils, and spherical

With four coils, also known as a double Helmholtz coil or double-pair Helmholtz coil, they can be also distributed over a sphere, further increasing the uniformity.23)

The positioning is such that the first pair is located at the intersection of 40.09° and the second pair at 73.43° angle with the outline of the sphere, as illustrated. Additionally, the turn ratio should be N1/N2 = 0.68212, the radius ratio r1/r2 = 0.67189, and the distance ratio q1/q2 = 2.685.24)

For an ordinary Helmholtz pair the optimal angle is 63.4°.25)

With an appropriate arrangement of turns and current intensity it is possible to achieve very uniform field in a spherical coil.26) However, such a shape there is no access to the inside of the coil and thus the practical applicability is highly limited. On the other hand, spherical coils can be used as sensors, because they detect magnetic field which is proportional to the field at the geometrical centre of such coil.27)

There are many other configuration with multiple coils which are provided commercially, both with different dimensions of sub-coils as well as those resembling solenoidal arrangement (with the same radius for all sub-coils).28) They can be used for generating magnetic field with specific non-uniformity, such as in the Penning trap.29)

### Other shapes

Large coils were also used for studies of magnetic navigation in birds. In such biological studies typically the generated field does not have to be as precise (as far as magnitude is concerned), so the coils can be made less precisely (in the included picture, some “bowing” of the wires can be seen at the top of the octagonal coil).

Large octagonal Helmholtz coil used by Prof. Wolfgang Wiltschko (pictured) at Frankfurt University for studying magnetic navigation in birds by Gerbil, CC-BY-SA-3.0

It is theoretically possible to create a triangular Helmholtz pair.30)31) However, such configuration will have reduced uniformity at the centre so it is not used in practice.

Analytical equations can be derived also for other shapes such as pentagonal, and can be even generalised to any polygonal current loops (between 3 and 15 sides, and more), by using the Taylor series expansion. The calculations can be performed for both ordinary, and anti-Helmholtz configuration.32)

The octagonal Helmholtz coil (pictured) will have field intensity somewhere between the square and the round coil, depending on the length of the sides.33)

Other, very complex shapes were studied by some researchers. For example, Thabius et al.34) investigated the theoretical possibility of a single coil (which is a single coil rather than a typical Helmholtz coil), but the aim was to achieve the best field uniformity for as large volume as possible.

The optimisation of the shape of the coil was carried out by an automated finite-element modelling, which resulted in a rather complex shape as shown in the illustration below. This shape cannot be described by an analytical equation, because it was not derived in an analytical way, but obtained directly by numerical means.

Inside the volume of the optimised coil the field uniformity is much better than for the ordinary Helmholtz coil. The assumption for the optimisation was that the current distribution within the cross-section of the coil conductor was uniform. The shape of such coil somewhat resembles a spherical coil described above.

Optimised coil for generation of uniform field proposed by Thabuis et al.:35) (a/c) 3D cut view of the reference Helmholtz coils and the optimised shape, (b/d) magnetic flux density B distribution in upper half of the 2-D revolved domain for the ordinary Helmholtz coils and the optimised shape, respectively. A. Thabuis, X. Ren, T. Duong, Y. Perriard, CC-BY-SA-4.0

## Field uniformity

Distribution of magnetic field strength H in a Helmholtz coil (cross-section view) The field is most uniform around the centre of the coil. In the first approximation, the following error region can be assumed within a spherical volume with radius r' (for a coil with diameter d = 2·r).

Error range Spherical volume with radius r'
±0.01% r' = 0.05·d
±0.1% r' = 0.10·d
±1% r' = 0.15·d
±5% r' = 0.25·d

Close to the coils the field is significantly higher, and outside of the coils it decays quickly with the distance.

Uniformity of field inside a round Helmholtz coil, normalised to the value at the centre; the contours have the same shape for coils of various sizes Uniformity of field inside a round Helmholtz coil: 0.1%, 1% and 5% error region vs. a sphere with radius r' = 0.1·d, 0.15·d and 0.25·d, respectively Along the main axis (and perpendicular to it), the highest field is at the centre, and it reduces away from it. However, along the diagonals the field increases because of the proximity to the coils, and is relatively very high close to the surface of the coils.

It is possible to slightly improve the range of uniformity by separating the coils by the distance of 1.01 of the radius. This lowers the field at the centre from 100% to 99.4% of the amplitude but improves the uniformity range, as pictured.

The magnitude of the field at the centre is roughly related to the spacing between the coils, so that the error of spacing (from the ideal) causes around half of that error in the magnitude. So for example, 1% of the spacing error means around 0.5% of magnitude error.

A spacing which is slightly too far is better for uniformity (as compared to the spacing which is slightly too close). This is relevant as far as the ± error is taken into account.

Uniformity of field inside a round Helmholtz coil, with spacing between the coils equal to the radius, and to 1.01 of the radius Uniformity of field inside a round Helmholtz coil, with the half-coils spaced too close or too far (90% or 110% was used to emphasize the type of changes) ## Generation of magnetic field

There is direct analytical relationship between the current in the Helmholtz coil and the generated magnetic field. Therefore, for a precisely made and/or calibrated coil it is sufficient to supply a known current, which can be done for example by a calibrated current source connected directly to the coil. It is also possible to use a calibrated ammeter to measure the current flowing in the coil and use the analytical equation (as shown above) to calculate the field present at the centre of the coil.

Helmholtz coils can be be used over the whole range from DC to high-frequency AC (even in MHz range and beyond). The upper limit is imposed by the voltage required to drive the inductance of the coil and the wavelength of the frequency.

The same approach can be used with a solenoid or a Maxwell coil.

### Series and parallel

For DC and low frequencies the default configuration for the Helmholtz coil is typically in series. This ensures that precisely the same current flows through both half-coils, as generated by the power supply. For this reason, a calibrated power supply can be used to drive directly the required magnitude of current, without the need to additionally measure that current.

However, the half-coils can be also connected in parallel and the current supplied by the source will be twice the current flowing in each half-coil. This could lead to a 50% error in the current magnitude, if the total current rather than the half-coil current is used in the calculations.

Parallel connection is more useful for high-frequency cases because it lowers the effective impedance of such Helmholtz coil and thus lower voltage is required to achieve the same level of magnetic field.

### High-frequency implications

Helmholtz coils can be used to very high frequencies, even in MHz range. However, there are certain limitations which needs to be taken into account:

• Inductance of the coils will result in impedance increasing with the frequency, which in turn will demand higher voltage in order to maintain the required current.
• Resonance conditions can be used to reduce the amount of reactive power that has to be delivered to the coil at a given frequency. An external capacitor can be used so that effectively an LC tank is created. The resonance can be achieved in series or parallel connection, depending on the requirements in a given circuit.
• Transmission line effects are related to the physical size of the Helmholtz coil, with the rule-of-thumb limit being wavelength at the given frequency of around 10 times longer than the coil size.
• Eddy currents can be induced in the supporting structure as well as in the copper of the wire, which can significant affect the field uniformity. Some commercially available Helmholtz coils are made on solid aluminium support, and these might not be suitable for higher frequencies.

For a series connection, inductance of a Helmholtz coil can be estimated from the following equation (7).36)37)38) However, the formula is only approximate39) so only an order-of-magnitude estimation can be made with it. There are several other equations given in the literature, but they are also estimations at best.40)41)

The mutual inductance between the half coils is around 0.11 of the inductance of each half coil.42)

(7) Inductance of Helmholtz coil (connected in series) 43)44)45)
$$L = N_{each}^2 · r · μ_0 · \left( \text{ln} \left(\frac{16·r}{d_b} \right) -2 \right)$$ (H)
where: $N_{each}$ - number of turns in each half-coil (unitless), $r$ - radius of the coils, $μ_0$ - permeability of vacuum (H/m), “ln” - natural logarithm function, $d_b$ - diameter of the bunch of wires in a coil (assuming that they are bunched up so that they form circular cross-section)

## Cancellation of magnetic field (3D coils)

It is possible to generate magnetic field in an arbitrary direction by using three Helmholtz coils positioned orthogonally to each other. Such 3D arrangement can be then used for cancellation of some external magnetic field, such as the Earth's field. This can be achieved for example by using a negative feedback from zero-field detector, so that the 3D coils are driven with appropriate currents to cancel out the field in question.46)

Once the field is cancelled a known field can be generated by yet another set of Helmholtz coils placed inside the outer coils, as shown in the photograph of the experiment.

Square coils are often used for such cancellation because they provide somewhat larger practical volume. For example, one of the photographs shows coils large enough so that a human subject can comfortably sit inside of it, with an arrangement that the geometrical centre of the coil coincides with the patient's head.

Some researchers also used Helmholtz coils (two or three sets) for generation of rotating magnetic field, for example for investigation of rotational magnetisation.47)

External 3D square Helmholtz coils used for compensation of Earth's magnetic field, and internal round Helmholtz coil used for generation of a known field (or a known field gradient) by Tomasz Kawalec, CC-BY-SA-4.0

## Anti-Helmholtz coil

Anti-Helmholtz coil has the same arrangement as normal Helmholtz coil, but the current in one of the half-coil is reversed The anti-Helmholtz coil or gradient Helmholtz coil (in some cases referred to as “Maxwell coil”) has the same or similar geometry, but the current direction is reversed in one of the half-coils, as if they are electrically connected in series opposition.48) This results in a distribution of magnetic field which is positive maximum near one half-coil, zero at the centre, and negative maximum near the other half-coil.

Effectively the fields from each half-coil subtract so with the same current the maximum values are lower than those for an ordinary Helmholtz coil.

At the centre of the coil the generated magnetic field has zero amplitude, but a well defined gradient. Such gradient coils are used in experiments which require precise gradient of magnetic field, rather than just uniform distribution.49)

The gradient can be calculated analytically, as a function of the distance between coils, as per equation (8).

(8) H gradient in anti-Helhmoltz coil50)
$$\frac{dH (x=0)}{dx} = \frac{3·I·N}{2}·\frac{r^2·q}{\left( r^2 + \frac{q^2}{4} \right)^{5/2}}$$ ( (A/m)/m ) ≡ (A/m2)
where: $I$ - current (A), $N$ - number of turns (unitless), $r$ - radius of half-coils (m), $q$ - distance between half-coils (m)

The maximum value of gradient is attained for $q = r$, which is for identical configuration to the ordinary Helmholtz coil (i.e. the distance between the half-coils is equal to their radius).

(9) Maximum value of the field gradient in anti-Helmholtz coil51)
$$\frac{dH (x=0)}{dx} = \frac{3·I·N}{2·\left(\frac{5}{4}\right)^{5/2}·r^2} ≈ 0.8586501 · \frac{I·N}{r^2}$$ (A/m2)

However, the linearity of the gradient can be improved (at the expense of the magnitude) by placing the half-coils at the distance q = √3·r, which also significantly increases the available volume between the coils where the linear gradient is generated.52)

(10) Improved homogeneity of the field gradient in anti-Helmholtz coil53)
$$\frac{dH}{dx} \left(x=0 \right) = \frac{3·\sqrt{3}·I·N}{2·\left(\frac{7}{4}\right)^{5/2}·r^2} ≈ 0.64129339 · \frac{I·N}{r^2}$$ (A/m2)
Field along the axis of a round anti-Helmholtz coil, shown as a sum of the field produced by each half-coil (assumed to be a “thin” current loop); r - radius of the half-coils, q - spacing between the half-coils ## Measurement of magnetic moment

Helmholtz coil can be also used for sensing of the magnetic field, rather than for its generation.

A commonly used configuration is to employ the Helmholtz coil to measure the magnetisation of permanent magnets, by means of detecting the magnetic moment.54) If the magnet size is “small” compared to the Helmholtz coil, then such magnet can be treated as a point-like magnetic dipole moment, for which at large distance the magnetic field does not depend on the shape of the magnetic dipole, but only on its magnitude and direction.55)

The magnet to be measured is placed at the centre of the coil, such that its axis of magnetisation coincides with the axis of the Helmholtz coil, and then the magnet is withdrawn along the axis (such method is sometimes referred to as the ballistic method), as defined by the international standard IEC 60404-14.56) It is also possible to perform the measurement by withdrawing the magnet towards the side (between the coils), rather than along the axis.57)

During the withdrawal, a voltage is induced in the Helmholtz coil due to changes of the magnetic flux, and this can be detected by means of a fluxmeter (integrating the voltage by an analogue or ditigal methods).

The flux can be related to the magnetic moment of the magnet, which in turn it can be related to the magnetic polarisation J or magnetisation M of the magnet. The measurements are less accurate as compared to a hysteresisgraph method, but they are relatively easy to make, useful and reliable, with equipment which is less costly.58)

A Helmholtz coil with fixed dimensions has a constant factor relating the current in the coil and the magnetic field $H_0$ it generates at its centre (x=0). This can be calculated as the ratio $k_H = H_0/I$, which for an ideal Helmholtz coil (infinitely thin, spaced by a distance equal to the radius, see also equation (1)), can be derived as in eq. (11), so there is a proportionality of 0.71554 (which is used by its reciprocal 1/kH = 1.39754, as in eq. (12)).59)60)61)

The measurement can be also performed by rotating the magnet (so that the locations of its poles N and S are swapped), but this induces twice the voltage, so the final result must be divided by a factor of 2.62)

(11) Helmholtz coil constant kH
$$k_H = \frac{H_0}{I} ≈ 0.71554·\frac{N_{each}}{r}$$ ( (A/m) / A) ≡ (1/m)
where: $H_0$ - magnetic field strength (A/m) at the centre of the Helmholtz coil, $I$ - current (A), $N_{each}$ - number of turns in each half-coil (unitless), $r$ - radius of the Helmholtz coil (m)
(12) Measurement of magnetic moment and magnetic polarisation J or magnetisation M 63)64)65)
(12a) $$J = \frac{j}{V_m} = \frac{\int{V dt}}{k_H·V_m} = \frac{ΔΦ}{k_H·V_m}$$ (T)
(12b) $$M = \frac{j}{μ_0·V_m} = \frac{\int{V dt}}{k_H·μ_0·V_m} = \frac{ΔΦ}{k_H·μ_0·V_m}$$ (A/m)
where: $J$ - magnetic polarisation (T) of the magnet, $j$ - magnetic dipole moment related to polarisation expressed in (Wb·m), $V_m$ - volume (m3) of the magnet, $\int{V dt}$ - integral (V·s) of the induced voltage $V$ (V), $ΔΦ$ - change of magnetic flux from the central position (zero) to “infinity” (sufficiently far away), $k_H$ - Helmholtz coil constant (1/m).

## Calibration

The analytical equations for Helmholtz coils can be calculated precisely, but physical systems cannot be built with an infinite precision. For this reason, even very complex calculations which attempt to include as many factors as possible (such as insulation thickness, and spacing between the wires) cannot completely represent the actual behaviour of a physical Helmholtz coil.66)67)

For generation of magnetic field, a calibrated sensor should be placed at the centre of the coil and the relationship between the current in the coil and the actual magnetic field can be calculated, and expressed as the coil constant.

For detection of magnetic field, a known source of magnetic field is required, such as magnetic dipole in a form of a pure nickel (99.99%) rod or sphere.68)69) 