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Let P0(t) be the probability of no count in time t. Then P0(t t) is the probability of no count in time t t. If the arrivals of the particles are assumed to be independent events, the probability of no count in time t t is the product of the probability of no count in time t and the probability of no count in time t. Therefore, neglecting terms involving ( t)2 and higher, we have (1) From (1) we obtain (2) i.e., dP0 dt Solving (3) by integration we obtain (3) ln P0 lP0 or dP0 P0 or P0(t) l dt lim P0(t

P0(t)[l P0(t)

CHAP. 14]

lP0(t)

Fig. 14-4

To determine c, note that if t 0, P0(0) c is the probability of no counts in time zero, which is of course 1. Thus c 1 and the required probability is (4) P0(t) e

One approach to clarifying the relationships is to introduce a di erent relationship called existence dependency (ED). Existence dependency relationships are de ned as follows: A class (parent) can be associated with a lower class (child) if the lower (child) class only exists when the upper (parent) class exists and each instance of the lower (child) class is associated with exactly one instance of the upper (parent) class. This relationship and inheritance can be used to represent any problem domain.

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Let P1(t) be the probability of 1 count in time t, so that P1(t t) is the probability of 1 count in time t Now we will have 1 count in time t t in the following two mutually exclusive cases: (i) 1 count in time t and 0 counts in time t (ii) 0 counts in time t and 1 count in time t The probability of (i) is P1(t)(1 t). The probability of (ii) is P0(t) t. Thus, apart from terms involving ( t)2 and higher, (1) This can be written (2) Taking the limit as (3) or (4) Multiplying by e t, this can be written (5) which yields on integrating (6) P1(t) te

around in the direction of the assumed current, the thumb points in the direction of the ux. Resulting positive directions for 1 and 2 are shown on the gure. If uxes 1 and 2 aid one another, then the signs on the voltages of mutual inductance are the same as the signs on the voltages of self-inductance. Thus, the plus sign would be written in all four equations (2) and (3). In Fig. 14-4, 1 and 2 oppose each other; consequently, the equations (2) and (3) would be written with the minus sign.

P1(t)(1 P1(t)

Natural Current Further understanding of coupled coils is achieved from consideration of a passive second loop as shown in Fig. 14-5. Source v1 drives a current i1 , with a corresponding ux 1 as shown. Now Lenz s law implies that the polarity of the induced voltage in the second circuit is such that if the circuit is completed, a current will pass through the second coil in such a direction as to create a ux opposing the main ux established by i1 . That is, when the switch is closed in Fig. 14-5, ux 2 will have the direction shown. The right-hand rule, with the thumb pointing in the direction of 2 , provides the direction of the natural current i2 . The induced voltage is the driving voltage for the second circuit, as suggested in Fig. 14-6; this voltage is present whether or not the circuit is closed. When the switch is closed, current i2 is established, with a positive direction as shown.

P0(t)

lP0(t)

Fig. 14-5

lP1(t)

EXAMPLE 14.2 Suppose the switch in the passive loop to be closed at an instant t 0 when i1 0. the sequence of the passive loop is (see Fig. 14-6). di2 di M 1 0 dt dt

Snoeck and Dedene. Existence Dependency: The Key to Semantic Integrity between Structural and Behavioral Aspects of Object Types. IEEE TOSE, April 1998.

If t 0, P1 (0) is the probability of 1 count in time 0, which is zero. Using this in (6), we find c2 Therefore, (7) P1(t) te

By continuing in this manner, we can show that the probability of exactly n counts in time t is given by (lt)n e lt (8) Pn(t) n! which is the Poisson distribution.

For t > 0,

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