Problem 31 - Fundamentals of Physics

Question

An electrical cable consists of 125 strands of fine wire, each having \(2.65 \mu \Omega\) resistance. The same potential difference is applied between the ends of all the strands and results in a total current of \(0.750\) A. (a) What is the current in each strand? (b) What is the applied potential difference? (c) What is the resistance of the cable?

Step-by-Step Solution

Answer

(a) 0.006 A; (b) 1.59 nV; (c) 21.2 nΩ.

Step 1: Understand the Problem

We have a cable with 125 wire strands. Each strand has a resistance of \(2.65\, \mu \Omega\). A total potential difference is applied, giving a total current of 0.750 A through the cable. We need to find individual current in each strand, the applied potential difference, and the overall resistance of the cable.

Step 2: Calculate Current in Each Strand

Since the same potential difference is applied across all strands, and each strand has the same resistance, the current through each strand can be calculated by dividing the total current by the number of strands. Thus, the current in each strand is \[ I_{strand} = \frac{0.750}{125} = 0.006\, \text{A} \].

Step 3: Calculate Applied Potential Difference

Using Ohm's Law \( V = IR \), we can calculate the potential difference applied across each strand. Each strand has a resistance of \(2.65\, \mu \Omega = 2.65 \times 10^{-6} \Omega\). Therefore, \[ V = 0.006 \times 2.65 \times 10^{-6} = 1.59 \times 10^{-8}\, \text{V} \].

Step 4: Calculate Total Resistance of the Cable

Since all strands are connected in parallel, the total resistance \( R_{total} \) of the cable is given by \[ \frac{1}{R_{total}} = 125 \times \frac{1}{2.65 \times 10^{-6}} \]. Solving this gives \[ R_{total} = \frac{2.65 \times 10^{-6}}{125} = 2.12 \times 10^{-8}\, \Omega \].

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle used to relate voltage, current, and resistance in electrical circuits. It can be mathematically expressed as \( V = IR \), where:

• \( V \) is the voltage (potential difference) across the circuit components,
• \( I \) is the current flowing through the circuit, measured in amperes (A), and
• \( R \) is the resistance of the circuit, measured in ohms (Ω).

Understanding this relationship is key when analyzing simple circuits, whether they are single pathways or complex networks.

In practice, when we know any two of the three variables—\( V \), \( I \), or \( R \)—we can determine the third. In our exercise, Ohm's Law was used twice: to find the potential difference across a single wire strand and to evaluate the overall characteristics of the cable when all strands work together. When applied to each strand, Ohm’s Law helped find the small voltage across each resistance using the calculated current for the strand and its known resistance.

Parallel Circuits

Parallel circuits have a unique property where components share the same voltage across their terminals. In a parallel arrangement, the total current flowing through the circuit is the sum of the currents flowing through each individual component. Each of the 125 strands in the cable operates like a separate pathway in a parallel circuit.

This means the same potential difference affects each strand equally. However, since each strand has its own distinct resistance, it can carry a specific amount of current independently. The exercise demonstrates this by splitting the total cable current into the number of strands, reflecting how current divides in a parallel setup.

To find the cable’s total resistance in a parallel circuit scenario, one must use the formula:

• \( \frac{1}{R_{total}} = \sum \frac{1}{R_i} \)

where \( R_i \) is the resistance of each strand. In this case, an important takeaway is that additional parallel paths decrease total resistance, effectively making it smaller than the resistance of individual strands.

Current Measurement

Current measurement is critical in analyzing electrical circuits, as it indicates how much charge flows through a conductor over a period of time. Measured in amperes (A), the current is often a focal point when assessing both safety and performance of electrical systems.

In the problem, the total current for the entire cable was given as 0.750 A. To understand its distribution in the parallel strand setup, it was essential to consider how the current would divide among the numerous paths, or strands. By dividing the total current by the number of paths (strands), we determined the individual strand current.

Tools like ammeters measure current, and it is typically inserted into the circuit to give a direct reading. During calculations, especially in parallel circuits, understanding how current flows and divides helps optimize design and ensure that no pathway exceeds its current limits, preventing potential hazards.