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DIFFERENTIAL AND INTEGRAL OPERATIONS
dx Figwt 2.6 Differential Volume
Notice, both p and v, are evaluated at (xo, yo, G ) in this expression. By defining the
vector 5 = p3, which is called the current density, Equation 2.43 can be written more
concisely as
(2.44)
This same type of calculation can be made for the top cross-hatched surface shown in Figure 2.6. Figure 2.7(b) shows that now a positive y carries the number of particles in the shaded region out of the volume. This side contributes
JNbottom ~-
- JJxo, yo, ~ 0 , t ) d x dY.
at
to the total d N / a t. Notice, in this case, we evaluate J, at the point (a. yo, zo + dz). Combining Equations 2.44 and 2.45 gives
dN0p -
- _ -Jz(x0, yo, zo + dz, t ) d x d y
at
aNbottom aNtop -
- [JZ(xo,yo, zo, t ) - Jz(xo,yo, zo + dz, t>] dx d y. (2.46)
d t at
dx dx (a> (b) Figure 2.7 Flow Across the Bottom and Top Surfaces
DIFFERENTIAL OPERATIONS 29
This last expression can be written in terms of the derivative of Jz, because in the differential limit we have
J z ( x o , Y O , zo + dz, t ) = Jz(xo, yo, zo? t ) + - (2.47)
Substitution of Equation 2.47 into Equation 2.46 gives
Working on the other four surfaces produces similar results. The total flow into the differential volume is therefore
which can be recognized as -V * J times d7. Combining this result with Equation 2.42 gives us the continuity equation:
(2.50)
For a positive divergence of J, more particles are leaving than entering a region so dp/dt is negative. This exercise provides us with a physical interpretation of the divergence. If the divergence of a vector field is positive in a region, this region is a source. Field lines are “born” in source regions. On the other hand, if the divergence in a region is negative, the region is a sink. Field lines “terminate” in sink regions. If the divergence of a vector field is zero in a region, then all field lines that enter it must also leave.
2.3.3 Physical Picture of the Curl The curl of a vector field is a vector, which describes on a local scale the field’s circulation. From the word curl, itself, it seems reasonable to conclude that if a vector field has a nonzero curl the field lines should be “curved,” while fields with zero curl should be “straight.” This is a common misconception. It is possible for vector field lines to appear as shown in Figure 2.8(a), clearly describing a “curved” situation, and have no curl. Also, the fields lines shown in Figure 2.8(b), which are definitely “straight,” can have a nonzero curl. To resolve this confusion, we must look at the curl on a differential scale. Consider a vector field v that is a function of only x and y. The curl of this field points in the z-direction, and according to Equation 2.29 is given by
(2.51)
for a Cartesian coordinate system.
