Question

Theintegral formof the continuity equation is written as: dqdt + ∮Sj.dS = Σdqdt + ∮Sj.dS = Σ where q is the amount of quantity in a certain volume V, t is the time, j is the flux of q per unit surface and per unit area, S is the surface, ∮S is the closed intergral over the surface of the volume V, and Σ is the sources / sinks of q. Now an example was given from the same source, to give an intuitive explanation, which included the surface integral along with sinks and sources, but it didn't include time derivative, so I'm curious to know how would the quantity (number of people) in the case of the following example would change with respect to time alone; the example is: In a simple example, V could be a building, and q could be the number of people in the building. The surface S would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when peopl e enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a source, Σ > 0), and decreases when someone in the building dies (a sink, Σ < 0). The example does include the time derivative term. When they say thatthe number of people in the building increasesthey are saying that dq/dt > 0 , for example. Maybe it becomes more clear if you first consider the source-free (Σ = 0) case, dqdt = −∮j⋅dS. Thenpeople enter the buildingcorresponds to the integral being negative1, so that dq/dt > 0, which in turn means thatthe number of people in the building increases with time. Similarly the integral being positive means people exit the building, and we get dq/dt < 0, or the number of people in the building decreases over time. You can make sense of the source term Σ in a similar way. Just note that the sources and sinks must be enclosed by the surface S (i.e. within the building) in order to count. The integration surface is oriented such thatoutwardsflux is positive. This is clear from context, even if it isn't explicitly stated. Time wasn't mentioned in the example, but it was still implicitly assumed.

E.g., in the line Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface) the example assumes that this increase happens over a time periodd td t over which the inward flux of people is j. Since the example has its focus onqq(the number of people), therefore the time interval has been integrated over.

Answer 1

The continuity equation is used to describe the conservation of a quantity in a given volume. In this example, it explains how the number of people in a building changes over time. The time derivative term indicates whether the number of people is increasing or decreasing.

Step-by-step explanation:

The continuity equation is used to describe the conservation of a quantity, such as mass or charge, in a given volume. In this example, the quantity q represents the number of people in a building and the surface S represents the walls, doors, roof, and foundation of the building.

The equation states that the number of people in the building increases when people enter (inward flux) and decreases when people exit (outward flux), and is also affected by sources (such as births) and sinks (such as deaths).

The time derivative term dq/dt represents the change in the number of people with respect to time, indicating whether the number of people is increasing or decreasing.