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close this bookIntroduction to Electrical Engineering - Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.)
close this folder5. Magnetic Field
View the document5.1. Magnetic Phenomena
View the document5.2. Force Actions in a Magnetic Field
Open this folder and view contents5.3. Electromagnetic Induction

5.2. Force Actions in a Magnetic Field

The forces occurring in a magnetic field, are utilised in many ways for the construction and operation of machines and devices. There are three different types of forces:

· force on boundary areas of substances having different permeabilities (e.g. iron - air)
· force on currents in a magnetic field
· force between two currents

On the basis of the field line pattern, one can always draw conclusions regarding the force actions when one imagines that the field lines are subject to two forces:

The field lines try hard to shorten themselves; this means that they are subjected to a longitudinal force. The field lines in addition try hard to get away from each other as far as possible; this means that they produce a lateral force exerted upon each other.

Force in a magnetic field at boundary areas

Boundary areas are surfaces at which the permeability changes by leaps and bounds, e.g. the boundary surfaces of a piece of iron. A piece of iron in air is attracted by a magnet due to the shortening tendency of the field lines (Fig. 5.8). This can be effected by a permanent magnet and also by an electromagnet.

Fig. 5.8. Force in a magnetic field at boundary areas

Electromagnets are preferred because they attain greater magnet flux densities. The force with which a ferromagnetic surface is drawn towards the non-ferromagneticum is

F/N = 40 (B/T)2 A/cm2



pull of one magnetic pole in N = newton


magnetic flux density in T


pole area in cm

In electromagnets, the magnetic field is produced by the electric current. This shows the advantage that the great forces can be controlled conveniently and from a place remote from the magnet by changing the current.

The basic form of all electromagnets is the coil with iron core and air gap (Fig. 5.9.). The part carrying the winding is called core (1), the connecting pieces in the ferromagnetic circuit yoke (2), and the movable ferromagnetic piece is termed as armature (3).

Fig. 5.9. Basic form of an electromagnet

1 - Core
2 - Yoke
3 - Armature

Electromagnets are used in large numbers in all fields of engineering. Here are a few characteristic examples:

- lifting magnets (see Fig. 5.10.); they are suitable for lifting iron parts having a large mass
- chucks for clamping and holding workpieces on machine tools
- electromagnetic clutches. They can be operated conveniently;

their power of coupling is adjustable even from larger distances without steps

- relays (Fig. 5.11.) for the electromagnetic control of switching actions in telecommunication and telecontrol engineering

- headphones in telephony

- electromagnetic loudspeakers

Fig. 5.10. Lifting magnet

1 - Steel casting
2 - Manganese steel (non-magnetic)
3 - Workpiece of iron

Fig. 5.11. Relay

1 - Winding
2 - Core
3 - Armature
4 - Working contacts
5 - Connecting receptacles for the working contacts

Example 5.2.

Calculate the force with which a relay armature is attracted when the magnetic flux density in the air gap is 0.5 T and the pole area 0.25 cm!


B = 0.5 T
A = 0.25 cm2

To be found:

F in N


F/N = 40 (B/T)2 A/cm2
F/N = 40 · 0.52 · 0.25
F/N = 40 · 0.25 · 0.25
F/N = 40 · 0.0625 · 0.25
F = 25 N

Force exerted upon currents in a magnetic field (electrodynamic force)

When in a homogeneous magnetic field, a current-carrying conductor is placed according to Fig. 5.12., a force is exerted on this conductor in the direction indicated by an arrow.

Fig. 5.12. Current-carrying conductor in a magnetic field

1 - Horseshoe magnet
2 - Movable current-carrying conductor
3 - North pole
4 - South pole

This phenomenon can be explained easily. The vortex field of the current-carrying conductor (hatched area in Fig. 5.13.) is superimposed on the present field (thin lines). In the directional conditions selected in the Figs. 5.12. and 5.13., a field weakening is obtained to the left of the conductor whereas a field strengthening is obtained to the right. The resultant field is represented by broad lines. From the force actions of the field lines (longitudinal force - lateria force), a force F perpendicular to the conductor in the direction of the lowest magnetic flux density is obtained. This force is has the greatest value when the conductor is perpendicular to the field because then the most intense densification of field line takes place. When reversing the current direction or the direction of the field, the force brought about will act to the opposite direction. When current and field direction are reversed at the same time, the force action remains in the same direction. For the direction of the action of force, the left-hand rule (motor rule) holds.

Fig. 5.13. Electrodynamic force

1 - North pole
2 - South pole

a) Superposition of the present homogeneous field on the field produced by the conductor

b) Resulting magnetic field

When extending the opened left hand into the magnetic field in such a way that the field lines enter the inner palm and the extended fingers point in the direction of the current flow, the thumb spread out indicates the force exerted on the conductor (Fig. 5.14.).

Fig. 5.14. Left-hand rule (motor rule)

The magnitude of force is derived from the electrodynamic law of forces

F = B · I · l




force exerted on the current-carrying conductor


magnetic flux density of the homogeneous magnetic field


current intensity in the conductor


length of the conductor added to the field

[F] = [B] [I] [l]
[F] = T · A · m = (V · s)/m2 · A · m
[F] = (V · A · s)/m = (W · s)/m and with (W · s)/m = N we have
[F] = N

Example 5.3.

Between the 20 cm wide poles of a magnet there is a magnetic flux density of 0.5 T. A conductor arranged vertically with respect to the field carries a current of 5 A. Calculate the force exerted on the conductor!


B = 0.5 T
I = 5 A
l = 20 cm

To be found:



F = B · I · l
F = 0.5 V · s/m2 · 5A · 0.2 m
F = 0.5 V · A · s/m = 0.5 W · s/m
F = 0.5 N

The electrodynamic force is remarkable and technically of greatest importance.

On the basis of the force action described above, the motor principle can be explained. A conductor loop or a coil is arranged in a homogeneous magnetic field (Fig. 5.15.).

Fig. 5.15. Current-carrying conductor loop in a magnetic field

1 - Magnet poles
2 - Pivoted conductor loop
3 - Sense of rotation
4 - North pole
5 - South pole

It is pivoted so that it can be rotated about its central axis which is perpendicular to the direction of the field. When a current of sufficient intensity passes through this coil, it will be subjected to a rotary motion the direction of which can be determined with the help of the left-hand rule. Since the force acting according to equation (5-11.) is exerted on adequate force arms (radius of the conductor loop), the torque according to equation (4.7.) is obtained which has been dealt with in Section 4.4. The motor principle is the basis of a series of electrical measuring instruments and electric motors.

Force between two currents

Between two parallel current-carrying conductors force actions are attained by the superposition of magnetic fields. The field patterns (Fig. 5.16.) show:

Fig. 5.16. Force between two currents

a) Currents flowing in the same direction

b) Currents flowing in opposite directions

Equidirectional currents attract themselves due to the shortening tendency of the field lines; currents of opposite direction repel each other due to the widening tendency of the field lines.

The forces occurring are small; they should be taken into account in case of high currents (e.g. short-circuits). Therefore, bus-bar systems, heavy-duty windings in transformers and current-limiting reactors must be mechanically stiffened and reliably fastened, taking a high safety factor into account.

In a magnetic field, different force actions occur. All of them are due to the shortening and widening tendencies of the field lines.

Force actions in a magnetic field at boundary surfaces are primarily caused by means of electromagnets. Decisive is the fact that the current, which passes through a coil, exert an attractive force on ferromagnetic bodies. This force is directed in such a way that the magnetic resistance is reduced. The value of the force is F ~ B2 and can be determined with the help of equation (5.10.).

Force actions on current-carrying conductors in a magnetic field result from the superposition of two magnetic fields. The direction of force can be determined with the help of the left-hand rule; its magnitude can be derived from equation (5.11.) (it can be considerable). The motor principle which can be derived, from these actions is the basis of the designing of electric motors and, thus, technically of greatest importance.

Forces also occur between current-carrying conductors, namely, an attractive force with equidirectional currents and a repelling force with currents flowing opposite to each other. But only with very intensive currents (e.g. short-circuit) these forces reach noticeable magnitudes which then have to be taken into consideration.

Questions and problems:

1. At the two poles of 15 mm x 20 mm each of an electromagnet, a magnetic flux density of 645 mT is present. Calculate the force with which the armature attached to the two poles is attracted!

2. Calculate the induction in front of the armature of a relay o which shows a pull of 2.5 N with a pole area of 0.25 cm2!

3. Explain why a magnetic field exerts a force on a current-carrying conductor and in which direction this fore acts!

4. Calculate the force with which a wire carrying a current of 20 A will turn aside; the wire runs through the 5 cm wide homogeneous magnetic field of a magnetic flux density of 850 mT at right angles!

5. Under which conditions occurs a) a repellent and b) an attracting force action exerted by two parallel current-carrying conductors? (substantiation!)