Question

State whether the divergence of the following vector fields is positive, negative, or zero: ( $a$ ) the thermal energy flow in $\mathrm{J}/({\mathrm{m}}^2-\mathrm{s})$ 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 $\mathrm{kg}/({\mathrm{m}}^2-\mathrm{s})$ 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.

Key concepts

Thermal Energy Flow

Thermal energy flow refers to the movement of heat from one place to another. In the context of a freezing ice cube, thermal energy moves from warmer surroundings to the colder ice cube. Imagine the ice cube absorbing heat like a sponge soaking up water. This movement occurs because heat naturally flows from warmer to cooler areas.

In physics, this flow is represented as a vector field, showing the direction and magnitude of heat transfer. The divergence of this vector field is the measure of how much the heat flow is spreading out at a particular point. When thermal energy is leaving the ice cube evenly from all points, it indicates that the vector field is expanding. An expanding vector field means the divergence is positive. This positive divergence is key in understanding how heat distributes in various materials, impacting everything from ice formation to climate patterns.

Current Density

Current density is an important concept when discussing how electric currents move through conductors. It represents the amount of electrical current flowing through a unit area of a conductor. Think of it like the number of cars traveling down a highway per kilometer. This is crucial in understanding how electricity is distributed in circuits.

In a bus bar carrying direct current, the current density remains constant across the conductor. This means that there is an uninterrupted flow of electrons from the positive to the negative terminal with no variation in magnitude or direction. In technical terms, this represents a vector field with zero divergence because the flow does not spread out or diminish at any point. Understanding this stability in current density helps in designing electrical systems to prevent overloading and ensure efficient energy distribution.

Mass Flow Rate

Mass flow rate is commonly associated with fluid dynamics and refers to the mass of fluid passing through a given surface area per unit time. Imagine pouring a large jug of water into a funnel, with the flow representing the mass of water moving through the funnel in a given timeframe.

In the case of water circulating below the surface in a basin, the direction of the flow changes, like dancers moving in a circular formation, but the magnitude remains constant. This phenomenon creates a vector field that is rotational without expansion or contraction. As a result, the divergence of the mass flow rate is zero, suggesting no net increase or decrease in the volume of water moving at any point. Such understanding of mass flow rates assists in designing efficient water distribution systems, from irrigation to city water supplies.

Vector Fields

Vector fields are mathematical representations used to illustrate how quantities change over space. They act like a map that displays both the direction and magnitude of a particular phenomenon at every point. For example, showing how winds circulate around the globe or how heat dissipates from a hot cup of coffee.

In physics and engineering, vector fields are used to describe a variety of phenomena, including thermal energy flow, current density, and mass flow rate. By understanding vector fields, scientists and engineers can predict behaviors such as how heat will spread in a building, ensuring efficient design that maximizes energy conservation. The concept of divergence then indicates whether a vector field is spreading out (positive divergence) or coming together (negative divergence), providing insights into the equilibrium or disequilibrium of a system.

Electromagnetics

Electromagnetics is the study of electric and magnetic fields, which are vector fields themselves. This field covers how these forces interact, move, and change over time. It's a cornerstone concept in physics that lays the groundwork for understanding modern technology such as communication devices, motors, and computers.

Electric fields arise from the movement of charges, while magnetic fields are generated by moving currents or magnetic materials. In applications like bus bars carrying direct current, electromagnetics provides the framework to understand why the current density remains constant and how electric currents are efficiently managed. The key takeaway from studying electromagnetics is how it combines with other principles, such as vector fields and divergence, to create a comprehensive understanding of how electric and magnetic forces operate and are applied in technology.