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 $${8 \over 3}\Omega $$.

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

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: