Calculating Equivalent Resistance of Parallel Wire Parts

Let's break down this problem about the resistance of a metallic wire when cut and connected in parallel. We are given a metallic wire with an initial resistance of 20 Ω.

Resistance of a wire is directly proportional to its length. This means if you cut a wire into equal parts, the resistance of each part will be a fraction of the original resistance, corresponding to the fraction of the length.

In this case, the wire is cut into two equal parts in length. So, the length of each new part is half the original length. Consequently, the resistance of each part will be half of the original resistance.

Original Resistance (\(R_{original}\)) = 20 Ω

Number of equal parts = 2

Resistance of each part (\(R_{part}\)) = \(\frac{R_{original}}{2} = \frac{20\, \Omega}{2} = 10\, \Omega\)

Now, these two parts, each having a resistance of 10 Ω, are connected in parallel. When resistors are connected in parallel, the equivalent resistance (\(R_{eq}\)) is calculated differently than when they are in series.

For two resistors (\(R_1\) and \(R_2\)) in parallel, the equivalent resistance is given by the formula:

\(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}\)

Alternatively, for exactly two resistors, you can use the product-over-sum formula:

\(R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}\)

In this problem, both parts have a resistance of 10 Ω. So, \(R_1 = 10\, \Omega\) and \(R_2 = 10\, \Omega\).

Using the product-over-sum formula:

\(R_{eq} = \frac{10\, \Omega \times 10\, \Omega}{10\, \Omega + 10\, \Omega}\)

\(R_{eq} = \frac{100\, \Omega^2}{20\, \Omega}\)

\(R_{eq} = 5\, \Omega\)

Using the reciprocal formula:

\(\frac{1}{R_{eq}} = \frac{1}{10\, \Omega} + \frac{1}{10\, \Omega}\)

\(\frac{1}{R_{eq}} = \frac{2}{10\, \Omega}\)

\(\frac{1}{R_{eq}} = \frac{1}{5\, \Omega}\)

Taking the reciprocal of both sides:

\(R_{eq} = 5\, \Omega\)

Thus, the resistance of the parallel combination of the two parts is 5 Ω.

Revision Table: Wire Resistance Calculation

Additional Information on Electrical Resistance

Electrical resistance is a measure of how much a material opposes the flow of electric current. It is measured in Ohms (Ω).

• Factors Affecting Resistance: The resistance of a uniform metallic wire depends on its length (L), cross-sectional area (A), and the resistivity (\(\rho\)) of the material. The relationship is given by \(R = \rho \frac{L}{A}\).
• Series Combination: When resistors are connected end-to-end in series, the total equivalent resistance is the sum of individual resistances: \(R_{series} = R_1 + R_2 + R_3 + ...\) The current is the same through each resistor.
• Parallel Combination: When resistors are connected across the same two points in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances: \(\frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...\) The voltage is the same across each resistor. The equivalent resistance in a parallel combination is always less than the smallest individual resistance.

Understanding how resistance changes with length and how resistors combine in series and parallel circuits is fundamental in electrical circuit analysis.