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current_loop

Current loop

Stan Zurek, Current loop, Encyclopedia Magnetica,
http://e-magnetica.pl/doku.php/current_loop

Current loop - a theoretical concept of a structure, consisting of a loop of conductor with an electric current.

Current loop
Magnetic field around a current loop

A circular loop of current is one of the simplest circuits which can generate well-defined magnetic field.1)

However, in some cases also different shapes of current loops are considered, e.g. rectangular.2)

The concept of loop of current can be also applied to a semi-classical approach of analysing behaviour of electrons in atoms.3)

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Magnetic field strength H of circular current loop (FEM calculation, cross-section view)
H along the axis of circular current loop
H along the radius of circular current loop

Magnetic field

The magnetic field strength H (scalar or vector) or magnetic flux density B at the centre of a circular loop (point P in the drawing) can be calculated for example by using the Biot-Savart law. In vacuum, the values are:4)

Magnetic field of current loop at its centre (point P)
$$ H_P(t) = \frac{I(t)}{2·r} $$ (A/m)
where: $I(t)$ - current (A) at time $t$ (s), $r$ - radius of the loop (m); assuming infinitely thin wire and a circular loop placed in a uniform medium

The direction of magnetic field follows the right-hand rule (if fingers are curled in the direction of current then the thumb shows the direction of magnetic field).

Along the axis of a circular loop the magnetic field can be calculated with the following equation.5)

Calculator of H along axis of circular current loop

This section is an interactive calculator.
Magnetic field strength H of a circular current loop along its axis
$$ H(t,x) = I(t) · \frac{1}{2}· \frac{r^2}{(x^2 + r^2)^{3/2}} $$ (A/m)
where: $I(t)$ - current (A) at time $t$ (s), $r$ - radius of the loop (m), $x$ - location (m) from the centre of the loop (the centre is located at point x = 0); assuming infinitely thin wire and a circular loop placed in a uniform medium

Current I =        Radius r =       

Position on axis x =

       H (A/m) =


At the centre of loop ($x$ = 0), the equation simplifies to the one included above. And at a large distance from the loop (such that $x \gg r$) the equation simplifies to $1/x^3$ proportionality, typical for a dipole for which the field intensity decreases with a cube of distance.

The value of magnetic field can be also calculated for a rectangular loop of current, as included below. Assuming the same current, compared to the circular loop, at the centre of a square loop (whose length of side is the same as the diameter of the circle) the magnetic field strength H is around 10% lower,6) because the corners of the square are farther from the centre. However, at a large distance along the loop axis, the field of the square loop is around 27% larger, because the magnetic moment is greater, due to the greater area (by a factor of 4/π).

Calculator of H along axis of rectangular current loop

This section is an interactive calculator.
Magnetic field strength H of rectangular current loop along its axis
$$ H(t,x) = I(t)· \frac{2}{π}·\frac{a·b}{\sqrt{4·x^2 + a^2 + b^2} }·\left(\frac{1}{4·x^2 + a^2} + \frac{1}{4·x^2 + b^2} \right) $$ (A/m)
where: $I(t)$ - current (A) at time $t$ (s), $a$ and $b$ - length of sides of the rectangular loop (m), $x$ - location (m) from the centre of the loop (the centre is located at point x = 0); assuming infinitely thin wire and a rectangular loop placed in a uniform medium; (use $a=b$ for a square loop)

Current I =        Side a =        Side b =

Position on axis x =               H (A/m) =


Magnetic moment

Magnetic moment of the current loop

The vectorial magnetic dipole moment of such loop is defined as the product of the current and the area of the loop:7)

$$ \vec m = I · A · \vec{a} $$ (A·m²) ≡ (J/T)
where: $I$ - current (A), $A$ - area of the loop (m2), $\vec{a}$ - unit vector normal to surface A

The direction of magnetic moment is also linked to the direction of current with the right-hand rule.

At a large distance from the dipole the shape of the loop is not important, and only the directions, current, and area matter.8)

Simplified illustration of the orbital magnetic moment $\vec m_o$ and spin magnetic moment $\vec m_s$ - in the semi-classical approximation the electron can be represented as a current loop

In a semi-classical approach, the orbital magnetic moment of an electron can be related to the equivalent electric current created by the moving charge.9)

The spin magnetic moment can be conceptually illustrated as the electron spinning around its own axis. The spin moment of electron is much stronger that the orbital moment.

However, the “orbit” and “spinning” analogies are not true, because the classical approach cannot be applied to quantum phenomena. And the sub-atomic particles inherently cannot be fully described by classical equations.10)

Source of current

When the current loop is used in analysis the current is assumed to be present, without consideration of how it was generated or how it is sustained.

However, a current loop can be created in reality, for example by using a loop of superconductor. The current can be induced in a driven mode and if the circuit is completed such that it is completely made of superconductor (no resistive parts) then current will flow infinitely long (remaining in a persistent mode). Such devices are used for generating ultra-stable magnetic field.11)

Also, a charge moving in a circular path, such as an electron orbiting an atomic nucleus effectively constitutes a current loop and a fundamental unit of magnetism.12) However, an electron moving on a circular path is accelerated (centripetally), which would lead to radiation of energy. This cannot be explained with a classical approach. Therefore, the “orbital electron movement” is just a crude analogy, and the behaviour can be described only by quantum phenomena.13)

Source of magnetic field

The concept of loops of current forms the basis for analysis of structures such as Helmholtz coil and solenoid (and similar).

Helmholtz coil - two circular coils separated by the distance equal to their radius
Solenoid is often used a source of known value of magnetic field strength H, which can be calculated for its geometrical centre (black circle)

A Helmholtz coil is essentially a pair of two circular current loops placed at a distance equal to their radius. Such arrangement allows generating a uniform magnetic field over relatively large volume inside the coil, and this is useful for scientific experiments.

It is also possible to make a Helmholtz coil with loops which have different shapes (e.g. square, octagonal, etc.) The field is less uniform, but extends over a larger volume.14)

A solenoid is essentially a series of circular loops placed at uniform intervals over some length. In a practical approach a solenoid can be wound as a helical coil of wire, with one or more layers.

A solenoid can typically have a much larger number of turns and thus can generate larger magnetic field than a Helmholtz coil.

Such (and other similar) arrangements which can generate uniform field can be used for calibration of other magnetic sensors, or for applications in which the uniformity is critical, as it is for example in magnetic resonance imaging.15)

See also

References

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current_loop.txt · Last modified: 2023/09/04 14:22 by stan_zurek

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