Page 101 - The Mechatronics Handbook

0066_Frame_C07 Page 11 Wednesday, January 9, 2002 3:39 PM

F +Q +Q F

Magnetic Force
Vector, F = I × B

Magnetic Field Vector
B

Electric
Current, I

FIGURE 7.10 Electric forces on two charges (top). Magnetic force on a current carrying wire element (bottom).

Electromagnetic systems can be modeled as either distributed field quantities, such as electric field E
or magnetic flux density B or as lumped element electric and magnetic circuits. The force on a point
charge Q is given by the vector equation (Fig. 7.10):

F = QE (7.22)

When E is generated by a single charge, the force between charges Q 1 and Q 2 is given by

F = ---------------- (7.23)
Q 1 Q 2
4pe 0 r 2

and is directed along the line connecting the two charges. Like charges repel and opposite charges attract
one another.
The magnetic force per unit length on a current element I is given by the cross product

F = I × B (7.24)

where the magnetic force is perpendicular to the plane of the current element and the magnetic field
vector. The total force on a closed circuit in a uniform field can be shown to be zero. Net forces on closed
circuits are produced by field gradients due to other current circuits or field sources.
Forces produced by field distributions around a volume containing electric charge or current can be
calculated using the field quantities of E, B directly using the concept of magnetic and electric stresses,
which was developed by Faraday and Maxwell. These electromagnetic stresses must be integrated over
an area surrounding the charge or current distribution. For example, a solid containing a current
2
distribution can experience a magnetic pressure, P = B t /2µ 0 , on the surface element and a magnetic
2
tension, t n = B n /2µ 0 , where the magnetic field components are written in terms of values tangential and
2
normal to the surface. Thus, a one-tesla magnetic field outside of a solid will experience 40 N/cm pressure
if the field is tangential to the surface.
In general there are four principal methods to calculate electric and magnetic forces:
• direct force vectors and moments between electric charges, currents, and dipoles;
• electric field-charge and magnetic field-current force vectors;

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