Question

In how many parts (equal) does a wire of $100\Omega $ be cut so that a resistance of $1\Omega $ is obtained by connecting them in parallel?
A) $10$
B) $5$
C) $100$
D) $50$

Answer 1

Hint: As length of the conductor increases, resistance of the conductor also increases and as area of cross section of the conductor increases then resistance decreases. Resistance is the opposition occurs because atoms and molecules of the substance obstruct the flow of charge carriers.

Complete step by step answer:
We know that the resistance of the conductor is directly proportional to the length of the conductor and inversely proportional to the area of cross section.
Then, $R \propto \dfrac{l}{A}$
After removing proportionality, we get, one constant that is $\rho $
$R = \rho \dfrac{l}{A}$
Where, $\rho $ is the resistivity of the conductor.
Now here, a resistance of a wire $100\Omega $ is divided into equal parts (half) so that we get total resistance as $1\Omega $ when it is connected in parallel. We need to find the total number of wires connected in parallel.
The resistance of one wire of length$l,$ $R = 100\Omega $ (original condition)
Now if we cut it in a half that is equal parts, then
$\dfrac{{{R_1}}}{1} = \dfrac{{{R_2}}}{{\dfrac{1}{2}}}$
Rearranging the above equation, then
${R_2} = \dfrac{{{R_1}}}{2}$
If we cut it in n- equal parts ten resistance will be,
${R_n} = \dfrac{R}{n}$…………(1)
Because, we shorten the length n-times so it became$\dfrac{1}{n}$
We have${R_p} = 1\Omega $ then each equal divided wires are connected in parallel, then we get
$\dfrac{1}{{{R_p}}} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + ........... + \dfrac{1}{{{R_n}}}$
In this problem all the resistance connected in parallel are having the same value of resistance. Then
$\dfrac{1}{{{R_p}}} = \dfrac{1}{R} + \dfrac{1}{R} + ........... + \dfrac{1}{R}$
$\dfrac{1}{1} = \dfrac{1}{R} + \dfrac{1}{R} + ........... + \dfrac{1}{R}$
$1 = \dfrac{n}{{{R_n}}}$
Substitute equation (1) in the above equation, we get
$1 = \dfrac{n}{{\dfrac{R}{n}}}$
$1 = \dfrac{{{n^2}}}{R}$
We have$R = 100\Omega $ then
$1 = \dfrac{{{n^2}}}{{100}}$
$ \Rightarrow n = 10$
$\therefore $ A resistance of $1ohm$is obtained by connecting 10 equal parts of the wire connecting them in parallel.

So, option a is correct

Note:Resistance of a conductor depends on the length, area of cross section, temperature of the conductor.
Resistance is the opposition offered by the substance to the flow of current.