Tensors for Physics

3.4 Applications of the Vector Product 45

wherewis an axial vector which is perpendicular to the plane of motion. The
magnitude of this vector is the angular velocityw. The pointr=0 is located on the
axis of rotation. For a circle with radiusR, the magnitude of the velocity isv=Rw.

3.4.4 Lorentz Force.

The forceFacting on a particle with chargee, moving with velocityv, in the presence
of an electric fieldEand a magnetic field (flux density)Bis

F=eE+ev×B. (3.47)

This expression is calledLorentz force. Notice thatF,Eandvare polar vectors,
with negative parity, whereasBis an axial vector with positive parity. Thus parity is
conserved in (3.47).
What about time reversal invariance? The electric fieldEis even under time
reversal, theB-field and the velocity are odd, i.e. they do change sign under time
reversal. Thus both terms on the right hand side of (3.47) do not change sign and the
same is true for the resulting force. Consequently, the Lorentz force conserves parity
and is time reversal invariant.

3.4.5 Screw Curve

The position vectors=s(α)describing a screw-like curve, as function of the angle
α(Fig.3.3), is given by

s=ρ(ecosα+usinα)+χ

α

2 π

e×u. (3.48)

Fig. 3.3 A screw curve for
ρ=1andχ= 1 /3. The unit
vectorsuandvare pointing
in thex-andy-directions.
The vertical line indicates
the axis of the screw

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