Question

A wire of resistance R is cut into 5 equal parts and connected in parallel. If the equivalent resistance of this combination is R', calculate the ratio R/R'​

Answer 1

AnsWer :

25 : 1.

To FinD :

Calculate the ratio R/R'

SolutioN :

Let,

• Resistance be R.
• Size of Wire A to B.

☛Condition :

• When We cutting a wire 5 Equal part its resistance become R/5 and When We connect in parallel combination it become 5/R.
• We have Formula for parallel combination → 1/R = 1/R1 + 1/R2 + 1/R3 + ..+ 1 /Rn.

★ Diagram.

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\put(0,0){\line(1,0){30}}\put(-3,0){ A }\put(31,0){ B }\put(1,-5){ \frac{R}{5} }\put(6,-5){ \frac{R}{5} } \put(12,-5){ \frac{R}{5} }\put(18,-5){ \frac{R}{5} }\put(24,-5){ \frac{R}{5} }\end{picture}$

★ Let's Find the ratio of R/R'.

$\tt \longmapsto\dfrac{1}{R'_Parallel }= \dfrac{5}{R} + \dfrac{5}{R} + \dfrac{5}{R} + \dfrac{5}{R} +\dfrac{5}{R}$

$\tt \longmapsto \dfrac{1}{R'} = \dfrac{25}{R}$

$\tt \longmapsto R' = \dfrac{R}{25}$

$\tt \longmapsto \dfrac{R}{R'} = 25.$

Therefore, the ratio of R/ R' is 25 : 1.

Answer 2

GiVeN :-

• A wire of resistance R is cut into 5 equal parts and connected in parallel.
• Equivalent resistance of this combination is R'.

To DeTeRmInE :-

The ratio R/R'.

AcKnOwLeDgEmEnT :-

$\bullet$ As the wire is cut into 5 equal parts, so the resistance of each part becomes R/5.

$\bullet$ For connections in parallel,

$\sf{\dfrac{1}{R_{Equivalent}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+...+\dfrac{1}{R_4} }$

SoLuTiOn :-

Putting the values :-

$\sf{\dfrac{1}{R_{Equivalent}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+...+\dfrac{1}{R_4} }$

$\displaystyle{\sf{\longmapsto \frac{1}{R'} =\frac{1}{R/5}+\frac{1}{R/5}+\frac{1}{R/5}+\frac{1}{R/5}+\frac{1}{R/5} }}\\\\\displaystyle{\sf{\longmapsto \frac{1}{R'}=\frac{5}{R}+\frac{5}{R} +\frac{5}{R} +\frac{5}{R} +\frac{5}{R} }}\\\\\displaystyle{\sf{\longmapsto \frac{1}{R'}=\frac{25}{R} }}\\\\\displaystyle{\sf{\longmapsto R'=\frac{R}{25} }}$

$\rule{120}2$

Now,

$\displaystyle{\sf{\frac{R}{R'}=\frac{R}{\frac{R}{25} } }}\\\\\\\displaystyle{\sf{\longmapsto \frac{R}{R'}=R \times \frac{25}{R} }}\\\\\\\boxed{\displaystyle{\sf{\longmapsto \frac{R}{R'}=\frac{25}{1} }}}$

Hence, the ratio of R/R' is 25 : 1.