Lorentz force
If a conductor carrying current is placed in a magnetic field, a force acts on it that is perpendicular to the conductor and the magnetic field. This force is called the Lorentz force. If the conductor is perpendicular to the magnetic field lines, the force is greatest.
Step-by-step explanation of the experimental setup shown in Figure 1 with a current-carrying conductor in the field of a permanent magnet:
- A conductor is located between the poles of a horseshoe magnet.
- The conductor is not electrically conductive here, so that the field of the permanent magnet can be considered individually.
- The magnetic field lines run from the north pole to the south pole outside the permanent magnet and close inside the magnet.
- Now the conductor is considered to be carrying current, without taking the field of the permanent magnet into account.
- A magnetic field is created around the current-carrying conductor depending on the direction of the current flow (right-hand rule).
- The field of the permanent magnet and the magnetic field of the current-carrying conductor are now considered together.
- Both fields overlap, resulting in a field amplification on the right side and a field attenuation on the left side.
- The system attempts to compensate for this field distortion, causing a force to act on the conductor that is perpendicular to the conductor and the magnetic field (three-finger rule).
- The conductor moves in the direction in which the force acts.
- Here, the direction of flow of the current-carrying conductor is reversed.
- The Lorentz force acts in the opposite direction (three-finger rule).
- The conductor moves in the direction in which the force acts.
The vector or scalar representation can be used to calculate the Lorentz force. In the vector calculation (equation ??), a moving unit charge \(q\) with velocity \(\vec {v}\) in the magnetic field \(\vec {B}\) causes the force \(\vec {F}\): \begin {align} \vec {F} = q \cdot (\vec {v} \cdot \vec {B}) \label {GlLorentzkraftvek} \end {align}
In the scalar calculation, the vectors are omitted and instead the angle sin \(\alpha \) in relation to the velocity \(v\) and the magnetic field \(B\) are included in the calculation. It applies that the Lorentz force is perpendicular to both the flux density and the velocity of the charge. The magnitude of the force is calculated for straight conductors and a constant magnetic field as follows: \begin {align} F &= q \cdot v \cdot B \cdot \sin {\alpha }\\ F &=I\cdot \ell \cdot N \cdot B\cdot \sin {\alpha }\label {GlLorentzkraft} \end {align}
Equation ?? illustrates the scalar calculation in a current-carrying conductor. \(\ell \) is the effective conductor length with the number of \(N\) parallel conductors carrying the current \(I\). \(\alpha \) is the angle between the conductor and the direction of the magnetic flux density \(\vec {B}\). If the conductor and the magnetic field are perpendicular to each other, \(\sin {\alpha }\) is omitted.
Key point: Lorentz force
The Lorentz force describes the force acting on a moving charge in a magnetic field. If the charge flows through a conductor, the Lorentz force acts perpendicular to both the current flow and the magnetic flux density.
To illustrate the principle of an electric motor, consider a current-carrying conductor loop that is movably attached to a rotary rotor in the field of a permanent magnet:
- A conductor loop is located on a rotating element, the rotor, in the magnetic field of a horseshoe magnet.
- No current is flowing through the conductor yet, so the field of the permanent magnet can be considered individually.
- The magnetic field lines run outside the permanent magnet from the north pole to the south pole.
- Now consider the current flowing through the conductor loop without taking the field of the permanent magnet into account.
- A magnetic field is created around the current-carrying conductor loop depending on the direction of the current flow (right-hand rule).
- The field of the permanent magnet and the magnetic field of the current-carrying conductor loop are now considered together.
- The Lorentz force, which is perpendicular to the conductor and the magnetic field, acts on the conductor loop.
- The conductor loop begins to rotate due to the Lorentz force.
- Here, the direction of flow of the current-carrying conductor is reversed.
- Accordingly, the polarity of the current-carrying conductor loop and thus the direction of rotation of the conductor loop are also reversed (three-finger rule).
Lorentzkraft A direct current motor has a magnetic flux density of \(B=0.8\,\mathrm {T}\) in the air gap. There are a total of \(N=400\) windings under the poles, through which a current of \(I=10\,\mathrm {A}\) flows. The effective conductor length is \(\ell =150\,\mathrm {mm}\).
Calculate the force \(F\) at the circumference of the anchor.\begin {align*} F & =B\cdot I\cdot \ell \cdot N \\ & =0,8\,\tfrac {\mathrm {Vs}}{\mathrm {m}^2}\cdot 10\,\mathrm {A}\cdot 0,15\,\mathrm {m}\cdot 400 \\ & =480\,\frac {\mathrm {kg}\cdot \mathrm {m}^2\cdot \mathrm {s}\cdot \mathrm {A}\cdot \mathrm {m}}{\mathrm {s}^3\cdot \mathrm {A}\cdot \mathrm {m}^2} = 480\,\mathrm {N} \end {align*}
Two parallel conductors through which current flows also exert a Lorentz force on each other, as each conductor through which current flows creates a magnetic field around itself (see Figure 2). If the direction of current is the same in both conductors, an attractive force is created; if the direction of current is reversed, the force is repulsive.
For the theoretical special case of two straight, parallel, thin and infinitely long wires, the following ratio applies:
\begin {equation} F_{1,2} = \frac {\ell \cdot \mu _0\cdot I_1\cdot I_2}{2\cdot \pi \cdot r} \end {equation}
Since these specific conditions do not apply in practice, the formula serves merely as an approximation of real cases in order to estimate the force exerted by the Lorentz force.
