Question 648: Current Electricity

Question

A ring is made of a wire having a resistance R
0
= 12 $\Omega $. Find the points A and B, as shown in the figure, at which a current carrying conductor should be connected so that the resistance R of the sub circuit between these point is equal to ${8 \over 3}\Omega $.
![](https://neetcare-image.cdn.examgoal.net/1qpqahkk7yfq9mc/6b6153cb-d4fb-485b-8034-893c672d27ee/8174f650-69b3-11ea-aedc-f1c8d8e0194b/file-1qpqahkk7yfq9md-384w.jpg)

Accepted Answer

Let x is the resistance per unit length then
equivalent resistance $R = {{{R_1}{R_2}} \over {{R_1} + {R_2}}}$
${{\left( {{x_1}{l_1}} ight)\left( {{x_2}{l_2}} ight)} \over {x{l_1} + x{l_2}}}$
$ \Rightarrow {8 \over 3} = x{{{l_1}} \over {{{{l_1}} \over {{l_2}}} + 1}}$   ...(i)
also $R_0 = xl_1 + xl_2$
$12 = x(l_1 + l_2)$
$12 = x{l_2}\left( {{{{l_1}} \over {{l_2}}} + 1} ight)$   ...(ii)
Now
${{\left( i ight)} \over {\left( {ii} ight)}} \Rightarrow {{{8 \over 3}} \over {{{12} \over 1}}} = {{{{x{l_1}} \over {\left( {{{{l_1}} \over {{l_2}}} + 1} ight)}}} \over {x{l_2}\left( {{{{l_1}} \over {{l_2}}} + 1} ight)}}$
$ = {{{l_1}} \over {{l_2}{{\left( {{{{l_1}} \over {{l_2}}} + 1} ight)}^2}}}$
${\left( {{{{l_1}} \over {{l_2}}} + 1} ight)^2} 	imes {8 \over {36}} = {{{l_1}} \over {{l_2}}}$
$\left( {{y^2} + 1 + 2y} ight) 	imes {8 \over {36}} = y$   (where y = ${{{{l_1}} \over {{l_2}}}}$)
8y
2
+ 8 + 16y = 36y
$ \Rightarrow $ 8y
2
Ã¢ÂÂ 20y + 8 = 0
$ \Rightarrow $ 2y
2
Ã¢ÂÂ 5y + 2 = 0
$ \Rightarrow $ 2y
2
Ã¢ÂÂ 4y Ã¢ÂÂ y + 2 = 0
$ \Rightarrow $  2y (y Ã¢ÂÂ 2) Ã¢ÂÂ 1(y Ã¢ÂÂ 2) = 0
$ \Rightarrow $  (2y Ã¢ÂÂ 1) (y Ã¢ÂÂ 2) = 0
$ \Rightarrow $ $y = {{{l_1}} \over {{l_2}}} = {1 \over 2}$ or 2
![](https://neetcare-image.cdn.examgoal.net/1qpqahkk7yfsks9/6579280c-d525-4bf8-b71d-fa126ecb9fcc/c1b2e290-69b3-11ea-aedc-f1c8d8e0194b/file-1qpqahkk7yfsksa-296w.jpg)

Answer

${{{l_1}} \over {{l_2}}} = {5 \over 8}$

Answer

${{{l_1}} \over {{l_2}}} = {1 \over 3}$

Answer

${{{l_1}} \over {{l_2}}} = {3 \over 8}$

Answer

${{{l_1}} \over {{l_2}}} = {1 \over 2}$

Current Electricity

A ring is made of a wire having a resistance R 0 = 12 \Omega . Find the points A and B, as shown in the figure, at which a current carrying conductor should be connected so that the resistance R of the sub circuit between these point is equal to {8 \over 3}\Omega .

Choose the correct answer: