Problem 31 - Fundamentals of Physics

Question

An electrical cable cons?sts of 125 strands of tine 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 in each strand, (b) 1.59 x 10^{-8} V applied potential, (c) Resistance is 2.12 x 10^{-8} Ω.

Step 1: Calculate current in each strand

To find the current in each strand, you need to divide the total current by the number of strands, as the total current is evenly distributed. Thus, the current in each strand is \( \frac{0.750 \text{ A}}{125} = 0.006 \text{ A} \).

Step 2: Calculate the applied potential difference

Ohm's law states \( V = IR \), where \( V \) is the potential difference, \( I \) is the current, and \( R \) is the resistance. With \( I = 0.006 \text{ A} \) and \( R = 2.65 \mu \Omega \), calculate the potential difference for each strand: \( V = 0.006 \times 2.65 \times 10^{-6} \Omega = 1.59 \times 10^{-8} \text{ V} \).

Step 3: Calculate the resistance of the cable

To find the equivalent resistance for strands in parallel, use the formula \( \frac{1}{R_{eq}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \cdots + \frac{1}{R_{n}} \). Since all strands have the same resistance \( R = 2.65 \mu \Omega \), this becomes \( \frac{1}{R_{eq}} = 125 \times \frac{1}{2.65 \times 10^{-6}} \). Thus, \( R_{eq} \approx 2.12 \times 10^{-8} \Omega \).

Step 4: Verify the results

Verify each step's result by checking the consistency. The total potential difference matches the calculated current and resistance: \( V = I \times R_{eq} = 0.750 \times 2.12 \times 10^{-8} = 1.59 \times 10^{-8} \text{ V} \), which confirms the solution is correct.

Key Concepts

Electric Current

Electric current refers to the flow of electric charge carried by moving electrons in a conductor, like a wire. To understand how current works, think of it as the rate at which electrons flow through a circuit. Just like water flows through a pipe, electric current flows through wires. The unit of electric current is the ampere (A), and it tells you how many electrons are passing through a point in the circuit each second.
In electric circuits, current is often split between multiple paths. When the current splits, each path or strand has an equal portion of the total current if the strands are identical. In our exercise, 125 strands carry a total current of 0.750 A, which means each strand carries the same small part of the current. By dividing the total current by the number of pathways (strands), you find that each one carries 0.006 A.

Resistance of a Wire

Resistance is a measure of how much a wire resists the flow of electric current. It is like a narrow or clogged section of a pipe that slows down the water flow. In electrical terms, the higher the resistance, the harder it is for current to flow. Resistance is measured in ohms (Ω).
Different factors affect resistance. For example, the material, length, and thickness of the wire play a role. Thin wires have more resistance than thick ones because there is less room for the electrons to move through. In the exercise, each strand of the cable has a resistance of 2.65 micro-ohms (\(2.65 \mu \Omega\)).
Ohm's Law, which is a core principle of electricity, relates resistance, voltage, and current by the formula \(V = IR\). This equation shows how much voltage pushes current through a given resistance. When you know the current and resistance, you can find the voltage across a wire wherever Ohm's Law applies.

Parallel Circuits

Parallel circuits are designed to give multiple pathways for electric current to flow. Think of parallel circuits like a river branching into different streams, where each stream can carry water independently. In the same way, multiple wires or strands in parallel will carry electric current independently of one another. This setup is common in household wiring, where each outlet or light is powered separately to ensure devices function even if others fail.
In a parallel circuit, the total resistance is reduced. This is because current divides across the multiple paths, lessening the overall obstruction to flow. The equivalent resistance \(R_{eq}\) in parallel, for equal resistors, is calculated using the formula\(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}\).
In the exercise, 125 strands with identical resistance each contribute to the overall reduction of the cable's resistance, resulting in an equivalent resistance lower than the resistance of a single strand. This feature of parallel circuits is why devices in these circuits are less affected when one part fails, as the current easily finds other routes to flow through.