(a) Using the data in Table \(18.1\), compute the resistance of a copper wire \(3 \mathrm{~mm}(0.12\) in.) in diameter and \(2 \mathrm{~m}(78.7 \mathrm{in}\).) long. (b) What would be the current flow if the potential drop across the ends of the wire is \(0.05 \mathrm{~V}\) ? (c) What is the current density? (d) What is the magnitude of the electric field across the ends of the wire?

Understanding copper wire resistance is crucial for anyone dealing with electronics or electrical systems. The resistance of a wire dictates how much it opposes the flow of electric current. Copper, being a commonly used conductive material, has a low resistance, making it an excellent choice for conducting electricity. To calculate resistance, one must consider the material's resistivity, the wire's length, and its cross-sectional area.

The formula R = \frac{\rho L}{A} (where R is resistance, \(\rho\) is resistivity, L is length, and A is area) is used to determine the resistance of a wire. Notably, longer wires or wires with smaller cross-sectional areas have higher resistance. In the provided exercise, we find the resistance of a particular copper wire by first calculating its cross-sectional area using the wire's diameter and then applying the formula with the given resistivity value.

Ohm's Law is a fundamental principle in the field of electricity and electronics, named after the physicist Georg Ohm. It describes the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit, succinctly defined by the formula \(V = IR\). This simple yet powerful law allows us to calculate one quantity if the other two are known. In our exercise, after determining the wire's resistance, we applied Ohm's Law to find out the current flowing through the copper wire when a potential difference of 0.05 V is applied across its ends.

Ohm's Law is not just a calculation tool but also a design guide. It is used to design circuits to manage current flow and ensure the safety and efficiency of electrical systems. As such, understanding and applying Ohm's Law is a foundational skill for anyone in the electrical engineering field.

Current density is a measure of the current flow per unit area across a cross-sectional area through which the current flows. It is denoted as J and expressed in amperes per square meter (A/m²). The formula to determine current density is \(J = \frac{I}{A}\), where I represents the current in amperes and A represents the cross-sectional area in square meters.

In practical terms, high current density can lead to overheating, which must be considered when designing electrical wires and components. In the context of the exercise, after calculating the current using Ohm's Law, the current density is found using the area of the wire computed in the earlier steps. It helps in understanding how densely packed the current is as it flows through the wire, an important consideration in designing electrical equipment.

The electric field is a fundamental concept in electromagnetism, representing the force exerted per unit charge at a point in space. Determining the strength of the electric field, denoted as E, involves the formula \(E = \frac{V}{L}\), with V being the potential difference or voltage and L being the length over which the voltage is applied. It is measured in volts per meter (V/m).

The electric field provides a view of the potential energy environment within an electrical circuit. It's critical when analyzing the force that would act on charged particles within the field. In the context of our exercise, calculating the electric field strength across the wire sheds light on how strong the driving force is that pushes the charges through the conductor. This piece of information is essential for a deeper understanding of circuit behaviors and for troubleshooting circuit-related issues.