Electromagnetism

A magnetic field can be generated not only by a permanent magnet, but also by an electric current. This results in a circular magnetic field around the current-carrying conductor. The direction of the field can be determined using the „right-hand rule“: When the hand is formed into a fist with the thumb extended, the thumb points in the direction of the (technical) electric current flow (the „technical current flow“runs from the positive to the negative pole) and the fingers point in the direction of the magnetic field lines.  

1 Magnetic flux

The magnetic fields generated by electric currents are measured using the parameter magnetic flux density \(\varTheta \) (theta). Analogous to electric voltage, flux density is also referred to as magnetic voltage. Since flux density results from current, it has the same unit as current, namely amperes.

The name is derived from the flux law. This states that the magnetic flux \(\varTheta \) is equal to the total current \(I\) of an area through which it flows (see Figure 2). Equation 1 represents the flux law in its general form. The right-hand side denotes the surface integral of the current density \(\vec {J}\). This integral expresses the sum of the current flowing through the surface \(A\). The left-hand side of the equation represents the closed line integral over the magnetic field strength \(\vec {H}\). This closed line \(\mathrm{d} \vec {s}\) corresponds to the boundary of the surface \(A\).

\begin {equation} \varTheta = \oint _s \vec {H}\cdot \mathrm{d} \vec {s} = \iint _A \vec {J}\,\mathrm{d} \vec {A}\label {Durchflutung} \end {equation}

Since in most cases the current is transported through a conductor and consequently both the direction and the strength of the current are clearly known, it is sufficient in this case to replace the integral (as shown in equation 2) with the number of current-carrying conductors \(N\) multiplied by the current strength \(I\). \begin {equation} \varTheta = \oint _s \vec {H}\cdot \mathrm{d} \vec {s} = N\cdot I\label {Durchflutungsgesetz} \qquad [\mathrm {A}] \end {equation}

If the magnetic field strength is constant over the integration path \(\mathrm{d} \vec {s}\), the vector \(\vec {H}\) can be moved outside the integral. This condition is usually fulfilled if the field line is located in the same material over the entire integration path. An example would be a circular field around a conductor at the centre of the circle or a toroidal coil, as shown in Figure 3. In this case, the integral \(\oint \vec {H}\cdot \,\mathrm{d} \vec {s}\) becomes the length of the integration path (Equation ??). This length is called the mean field line length and is denoted by \(\ell _{\mathrm {m}}\). It represents the mean value of the sum of all field lines located within the circular field. The magnetic field strength in such arrangements can be easily determined using equation ??.

\begin {align} \varTheta &= \oint _s \vec {H}\cdot \mathrm{d} \vec {s} = |\vec {H}| \cdot \ell _{\mathrm {m}}\label {GlmagnFeldstaerke}\\ |\vec {H}| &=\frac {\varTheta }{\ell _{\mathrm {m}}}\qquad \left [\frac {\mathrm {A}}{\mathrm {m}}\right ]\label {GlmagnFeldstaerke1} \end {align}

Magnetic field strength over the mean field line length A toroidal coil (Figure 3) with \(1000\) turns and an average field line length of \(50\,\mathrm {cm}\) is traversed by a current of \(100\,\mathrm {mA}\). How strong is the magnetic field?\begin {align} \varTheta &= H \cdot \ell _{\mathrm {m}} = N\cdot I\nonumber \\ H&=\frac {N\cdot I}{\ell _{\mathrm {m}}} =\frac {1000\cdot 0,1\,\mathrm {A}}{0,5\,\mathrm {m}} = 200\,\tfrac {\mathrm {A}}{\mathrm {m}}\nonumber \end {align}

Magnetic field strength across the circumference A direct current of \(I=50\,\mathrm {A}\) flows through a straight conductor. How strong is the magnetic field at a distance of \(r=20\,\mathrm {cm}\)? \begin {equation*} H=\frac {N\cdot I}{\ell _{\mathrm {m}}} =\frac {1\cdot 50\,\mathrm {A}}{2\pi \cdot 0,2\,\mathrm {m}} = 39,79\,\tfrac {\mathrm {A}}{\mathrm {m}} \end {equation*} The mean field line length \(\ell _{\mathrm {m}}\) can be determined by the circumference of the circle and is therefore replaced by the formula for calculating the circumference of a circle, \(2\pi \cdot r\).

2 Magnetic flux and flux density

Magnetic flux is comparable to electric current in an electric circuit. Contrary to the terminology, however, there is no flow of magnetic particles; rather, it is symbolically understood as „the amount of magnetic field“ and acts as a result of magnetic tension. The symbol for magnetic flux is \(\varPhi \), and the unit is the weber (Wb). One weber (Wb) is equivalent to one volt-second (Vs).

Apart from the magnetic flux, the force exerted by a magnet also depends on the area through which the flux passes. The more concentrated the field lines are, the greater the magnetic effect. This is described by the magnetic flux density \(B\), which in the simplest case (non-curved surface, homogeneous flux density) is defined by the quotient of the magnetic flux \(\varPhi \) and the area \(A\). The unit of flux density is Tesla (T). The direction of the flux density \(\vec {B}\) is perpendicular to the area, which is expressed by the normal vector to the area \(\vec {A}\).

\begin {equation} \vec {B} = \frac {\varPhi }{\vec {A}}\qquad [\mathrm {T}] \end {equation}

In general (without the above restrictions), the following applies:

\begin {equation} \varPhi = \iint _A \vec {B}\cdot \mathrm {d}\vec {A} \label {magnFlussFormel} \end {equation}

Both magnetic field strength and magnetic flux density are vector quantities. They can therefore be represented graphically by field lines. Magnetic flux \(\varPhi \), on the other hand, is a scalar quantity.

The magnetic flux density \(\vec {B}\) and the magnetic field strength \(\vec {H}\) are linked via the permeability \(\mu \). Permeability consists of the product of a material-independent parameter, the magnetic field constant \(\mu _0=1.256\,637\,062\cdot 10^{-6}\,\frac {\mathrm {Vs}}{\mathrm {Am}}\), and a material-specific permeability \(\mu _{\mathrm {r}}\). The magnetic field constant describes the permeability in a vacuum and, until the reorganisation of the SI units in 2019, was precisely defined with the value \(\mu _0 = 4\pi \cdot 10^{-7}\,\frac {\mathrm {Vs}}{\mathrm {Am}}\). It is now subject to measurement uncertainty.

\begin {equation} \vec {B} = \mu _0\cdot \mu _{\mathrm {r}}\cdot \vec {H}\label {GlFlussdichte} \end {equation} In a ferromagnetic material, the relationship between the magnetic field strength \(\vec {H}\) and the magnetic flux density \(\vec {B}\) is not linear. The permeability reaches saturation as the magnetisation increases, so that the relative permeability \(\mu _{\mathrm {r}}\) approaches 1 from a material-dependent initial value. If the magnetic field is reduced again (or set to zero), the magnetisation remains to a certain extent (point \(B_{\mathrm {r}}\) in Figure ??). This process is called remanence. The remaining magnetic flux density at a magnetic field strength of zero is the remanence flux density \(B_{\mathrm {r}}\). To completely demagnetise the material again, an inverse magnetic field strength, the coercive field strength \(H_{\mathrm {c}}\), is required.

Ferromagnetic materials are divided into hard and soft magnetic materials based on their coercive field strength. Hard magnetic materials (e.g. strong permanent magnets made of neodymium- iron-boron) have a value for \(H_{\mathrm {c}}\) greater than \(10\cdot 10^{3}\,\ frac{\mathrm {A}}{\mathrm {m}}\), while for soft magnetic materials (e.g. magnetic cores made of manganese-zinc ferrite), \(H_{\mathrm {c}}\) is less than \(500\,\frac {\mathrm {A}}{\mathrm {m}}\). Hard magnetic materials are mainly used for permanent magnets.

Magnetic flux density Inside a tightly wound toroidal coil, a magnetic field strength of \(H=100\,\frac {\mathrm {A}}{\mathrm {m}}\) is to be generated. The coil has a mean radius of \(5\,\mathrm {cm}\).

1.

Calculate the required current \(I\) if the coil has \(N=200\) windings.

From equation 2 and ??: \begin {align*} \varTheta & = H \cdot \ell _{\mathrm {m}} = N\cdot I \\ I & =\frac {H\cdot \ell _{\mathrm {m}}}{N}=\frac {100\,\frac {\mathrm {A}}{\mathrm {m}}\cdot 2\cdot \pi \cdot 5\cdot 10^{-2}\,\mathrm {m}}{200} = 157,08\,\mathrm {mA} \end {align*}

2.

How large will the flux density \(B\) be in the case of an air coil (\(\mu _{\mathrm {r}}=1\)) or an iron-filled coil (\(\mu _{\mathrm {r}}=2000\) at the operating point)?

From equation 5: \begin {align*} B_\mathrm {Luft} & = \mu _0\cdot \mu _{\mathrm {r}}\cdot H = 1,256\cdot 10^{-6}\,\tfrac {\mathrm {Vs}}{\mathrm {Am}} \cdot 100\,\tfrac {\mathrm {A}}{\mathrm {m}} = 125,6\,\mu \mathrm {T} \\ B_\mathrm {Eisen} & = 2000\cdot B_\mathrm {Luft} = 251,2\,\mathrm {mT} \end {align*}