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we can separately do the exponentiation of the third component giving e = 1;
the exponentiation of the top block can be performed along the same steps,
using the Cayley-Hamilton techniques from Example 8.12 , giving finally:

 cos(α t) sin(α t)  0

e A t =− sin(α t) cos(α t) 0 
 
 0 0  1

Therefore, we can write the solutions for the electron’s velocity components
as follows:

 vt()  cos(α t) sin(α t) 0  v 0() 0  
1
1
 vt() =− sin(α cos(α   v 0() + β  



0
  2    t) t) 0    2    
 vt()  0 0 1  v 0() t  
3
3
or equivalently:
vt( ) = v 0( )cos(α t) + v 0( )sin(α t)
1 1 2
vt( ) =− v 0( )sin(α t) + v 0( )cos(α t)
2 1 2
vt() = v 0( ) + β t
3 3

In-Class Exercises

Pb. 8.20 Plot the 3-D curve, with time as parameter, for the tip of the velocity
5
vector of an electron with an initial velocity v = v ê , where v = 10 m/s, enter-
0 1
0
ing a region of space where a constant electric field and a constant magnetic
r
4
flux density are present and are described by: E = E ê , where E = –10 V/m,
r
0 3
0
2
–2
and B = B ê , where B = 10 Wb/m . The mass of the electron is m = 9.1094
0 3
0
e
–31
× 10 kg, and the magnitude of the electron charge is e = 1.6022 × 10 –19 C.
Pb. 8.21 Integrate the expression of the velocity vector in Pb. 8.20 to find the
parametric equations of the electron position vector for the preceding prob-
lem configuration, and plot its 3-D curve. Let the origin of the axis be fixed to
where the electron enters the region of the electric and magnetic fields.
Pb. 8.22 Find the parametric equations for the electron velocity if the electric
field and the magnetic flux density are still parallel, the magnetic flux density
r
is still constant, but the electric field is now described by E = E cos(ωt)ê .
0
3
© 2001 by CRC Press LLC