Calculating Resistivity Of A Metal Rod A Step-by-Step Guide

In the realm of physics, understanding the electrical properties of materials is crucial for various applications, from designing electronic circuits to analyzing the behavior of conductors. One fundamental property is resistivity, which quantifies a material's ability to resist the flow of electric current. This article delves into the concept of resistivity and provides a step-by-step guide on how to calculate it, using a practical example of a metal rod.

Understanding Resistivity: The Key to Material's Electrical Behavior

In the realm of electrical conductivity, resistivity emerges as a crucial material property, acting as a measure of a material's opposition to the flow of electric current. Imagine electrons as tiny messengers carrying electrical signals through a material. In a material with high resistivity, these messengers encounter numerous obstacles, hindering their movement and impeding the flow of current. Conversely, materials with low resistivity offer a smooth pathway for electrons, facilitating the effortless transmission of electrical signals.

Resistivity, often denoted by the Greek letter rho (ρ), is an intrinsic property of a material, meaning it remains constant regardless of the material's shape or size. It is typically expressed in ohm-meters (Ω⋅m), reflecting its dependence on both resistance and geometry. Think of resistivity as a material's electrical fingerprint, uniquely characterizing its ability to conduct electricity.

To grasp the significance of resistivity, let's delve into its relationship with other electrical concepts. Resistance (R), measured in ohms (Ω), quantifies a specific object's opposition to current flow. Unlike resistivity, resistance depends on the object's dimensions and material. A long, thin wire will exhibit higher resistance than a short, thick wire made of the same material. This distinction highlights the importance of resistivity as a material-specific property, independent of geometry.

The connection between resistance, resistivity, length (L), and cross-sectional area (A) is elegantly captured by the formula:

R = ρL/A

This equation reveals that resistance is directly proportional to resistivity and length, while inversely proportional to cross-sectional area. A material with high resistivity will naturally exhibit higher resistance, and a longer object will offer more resistance than a shorter one. Conversely, a larger cross-sectional area provides more pathways for current flow, reducing resistance.

Resistivity plays a pivotal role in numerous applications. In electrical wiring, materials with low resistivity, such as copper and aluminum, are preferred to minimize energy loss during transmission. These materials act as excellent conductors, allowing electricity to flow freely with minimal resistance. Conversely, materials with high resistivity, such as rubber and glass, are employed as insulators to prevent current leakage and ensure electrical safety. Insulators act as barriers, effectively blocking the flow of electricity.

Semiconductors, like silicon and germanium, occupy a fascinating middle ground in the resistivity spectrum. Their resistivity can be precisely controlled by introducing impurities, making them essential components in transistors and integrated circuits. By carefully manipulating the resistivity of semiconductors, engineers can create electronic devices with a wide range of functionalities.

In summary, resistivity stands as a cornerstone concept in understanding a material's electrical behavior. It serves as a unique identifier, quantifying a material's inherent opposition to current flow. Its influence extends across diverse applications, shaping the design and functionality of electrical circuits, wiring systems, and electronic devices.

Problem Statement: Finding the Resistivity of a Metal Rod

Let's consider a practical problem to illustrate the calculation of resistivity. Imagine a metal rod with a length of 0.20 meters and a radius of 0.01 meters. When a voltage is applied across the rod, it exhibits a resistance of 3.2 x 10⁻⁵ ohms. Our objective is to determine the resistivity of the metal that constitutes the rod. This problem exemplifies how we can use measurable quantities like length, radius, and resistance to deduce a fundamental material property like resistivity.

Step 1: Gather the Known Information

Before embarking on the calculation, let's consolidate the information provided in the problem statement. This ensures we have all the necessary pieces of the puzzle at our fingertips.

• Length of the metal rod (L): 0.20 meters
• Radius of the metal rod (r): 0.01 meters
• Resistance of the metal rod (R): 3.2 x 10⁻⁵ ohms

Having these values clearly defined sets the stage for the subsequent steps in our resistivity calculation.

Step 2: Calculate the Cross-Sectional Area

In our resistivity equation, the cross-sectional area (A) plays a crucial role. For a cylindrical rod, the cross-sectional area is the area of the circular face, which can be calculated using the formula:

A = πr²

where π (pi) is a mathematical constant approximately equal to 3.14159, and r is the radius of the rod.

Plugging in the given radius (r = 0.01 meters) into the formula, we get:

A = π(0.01 m)²

A ≈ 3.14159 x (0.0001 m²)

A ≈ 3.14159 x 10⁻⁴ m²

Thus, the cross-sectional area of the metal rod is approximately 3.14159 x 10⁻⁴ square meters. This value represents the effective area through which electric current can flow within the rod.

Step 3: Apply the Resistivity Formula

With the resistance (R), length (L), and cross-sectional area (A) at our disposal, we can now invoke the fundamental resistivity formula:

ρ = RA/L

This equation elegantly connects resistivity (ρ) to resistance, cross-sectional area, and length. By rearranging the equation, we isolate resistivity as the subject, allowing us to calculate its value directly.

Substituting the known values into the formula, we have:

ρ = (3.2 x 10⁻⁵ Ω) x (3.14159 x 10⁻⁴ m²) / (0.20 m)

This step involves a straightforward substitution of numerical values, setting the stage for the final calculation.

Step 4: Perform the Calculation

Now, let's crunch the numbers to arrive at the resistivity value. Performing the multiplication and division in the equation, we obtain:

ρ ≈ (1.0053 x 10⁻⁸ Ω⋅m²) / (0.20 m)

ρ ≈ 5.0265 x 10⁻⁸ Ω⋅m

Therefore, the resistivity of the metal rod is approximately 5.0265 x 10⁻⁸ ohm-meters. This value represents the inherent resistance of the metal to the flow of electric current.

Step 5: Express the Answer with Correct Units

It is crucial to express the final answer with the appropriate units to ensure clarity and accuracy. In this case, resistivity is measured in ohm-meters (Ω⋅m), which reflects its definition as resistance per unit length per unit area.

Thus, the resistivity of the metal rod is approximately 5.0 x 10⁻⁸ Ω⋅m (rounded to two significant figures). This concludes our step-by-step calculation, providing a quantitative measure of the metal's resistivity.

Conclusion: Resistivity Unveiled

In this comprehensive guide, we embarked on a journey to understand and calculate the resistivity of a metal rod. We began by defining resistivity as a material's inherent opposition to electric current flow, emphasizing its importance in electrical conductivity. We then tackled a practical problem, meticulously outlining each step of the calculation process. By gathering known information, calculating the cross-sectional area, applying the resistivity formula, performing the calculation, and expressing the answer with correct units, we successfully determined the resistivity of the metal rod.

The calculated resistivity value of 5.0 x 10⁻⁸ Ω⋅m provides valuable insight into the metal's electrical behavior. This value can be compared to known resistivities of various metals to identify the material or assess its suitability for specific applications. For instance, metals with low resistivities, like copper and aluminum, are preferred for electrical wiring due to their ability to conduct electricity efficiently.

The concept of resistivity extends far beyond this specific example. It serves as a fundamental property in materials science, electrical engineering, and various other fields. Understanding resistivity allows us to design efficient electrical circuits, select appropriate materials for different applications, and gain deeper insights into the behavior of matter at the atomic level. As you delve further into the world of physics and electrical phenomena, resistivity will undoubtedly remain a crucial concept in your toolkit.

Answer the Question

Based on the calculation above, the resistivity of the metal is approximately 5.0 x 10⁻⁸ ohm-meters. Therefore, the correct answer is B) 5.0 x 10⁻⁸.