Question

State whether the divergence of the following vector fields is positive, negative, or zero: ( \(a\) ) the thermal energy flow in \(\mathrm{J} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\) at any point in a freezing ice cube; \((b)\) the current density in \(\mathrm{A} / \mathrm{m}^{2}\) in a bus bar carrying direct current; \((c)\) the mass flow rate in \(\mathrm{kg} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\) below the surface of water in a basin, in which the water is circulating clockwise as viewed from above.

Short answer

Answer: (a) Positive, (b) Zero, (c) Zero

Step-by-step solution

Observe the thermal energy flow

In a freezing ice cube, thermal energy flows from the warmer surroundings to the colder ice cube. This means that thermal energy is leaving the ice cube in all directions. The rate at which thermal energy leaves the ice cube is constant so the divergence of the flow is constant too.

Determine the divergence of the thermal energy flow

Since the thermal energy flows away from the ice cube and the flow is equally distributed, this indicates an expanding vector field, which means the divergence of the thermal energy flow is positive. #b) The current density in a bus bar carrying direct current#

Observe the current density in a bus bar

In a bus bar carrying direct current, the current flows in a straight line from the positive end to the negative end with no change in direction or magnitude. This means that the current density throughout the bus bar is constant and there are no differences in the flow of current at any point.

Determine the divergence of the current density in a bus bar

Since the current density is constant and doesn't spread out or converge at any point, the divergence of the current density in a bus bar carrying direct current is zero. #c) The mass flow rate below the surface of water in a basin circulating clockwise#

Observe the mass flow rate in the basin

In a basin with water circulating clockwise, the mass flow rate at any point below the water surface changes direction but remains constant in magnitude. This vector field presents a rotational behavior.

Determine the divergence of the mass flow rate in the basin

The fact that the water is circulating means that there isn't an expansion or convergence of the flow happening. Therefore, the divergence of the mass flow rate below the surface of water in a basin circulating clockwise is zero. In conclusion: 1. The divergence of the thermal energy flow in a freezing ice cube is positive. 2. The divergence of the current density in a bus bar carrying direct current is zero. 3. The divergence of the mass flow rate below the surface of water in a basin circulating clockwise is zero.