Question 395: Current Electricity

Question

A wire of resistance $$20 \Omega$$ is divided into 10 equal parts, resulting pairs. A combination of two parts are connected in parallel and so on. Now resulting pairs of parallel combination are connected in series. The equivalent resistance of final combination is _________ $$\Omega$$.

Accepted Answer

Let's start by understanding the process of dividing the wire and recombining its parts to form the final configuration. Initially, we have a wire with a resistance of $$20 \Omega$$. This wire is divided into 10 equal parts, each part then has a resistance of:
$$\frac{20 \Omega}{10} = 2 \Omega$$
Since each part has the same length and presumably the same material and cross-sectional area, then each part will have the same resistance of $$2 \Omega$$.
When two parts are connected in parallel, the equivalent resistance, $$R_{	ext{parallel}}$$, of this configuration can be calculated using the formula for two resistors in parallel:
$$\frac{1}{R_{	ext{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2}$$
Given that $$R_1 = R_2 = 2 \Omega$$ (since the parts are identical), we have:
$$\frac{1}{R_{	ext{parallel}}} = \frac{1}{2 \Omega} + \frac{1}{2 \Omega} = \frac{2}{2 \Omega}$$
This simplifies to:
$$\frac{1}{R_{	ext{parallel}}} = \frac{2}{2 \Omega} = \frac{1}{\Omega}$$
From which it follows that:
$$R_{	ext{parallel}} = 1 \Omega$$
Now, since the original wire was divided into 10 equal parts, and pairs of these parts are connected in parallel, this results in $$\frac{10}{2} = 5$$ pairs. Each of these pairs has an equivalent resistance of $$1 \Omega$$.
Finally, these pairs are all connected in series. The total resistance of resistors in series is simply the sum of their individual resistances. Therefore, the equivalent resistance of the final configuration, $$R_{	ext{series}}$$, is:
$$R_{	ext{series}} = 5 	imes R_{	ext{parallel}} = 5 	imes 1 \Omega = 5 \Omega$$
So, the equivalent resistance of the final combination is $$5 \Omega$$.

Current Electricity

A wire of resistance $20 \Omega is divided into 10 equal parts, resulting pairs. A combination of two parts are connected in parallel and so on. Now resulting pairs of parallel combination are connected in series. The equivalent resistance of final combination is _________ \Omega$.

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