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 points is equal to $\frac{8}{3} \Omega$

• A. $\frac{l_1}{l_2}=\frac{5}{8}$
• B. $\frac{l_1}{l_2}=\frac{1}{3}$
• C. $\frac{l_1}{l_2}=\frac{3}{8}$
• D. $\frac{l_1}{l_2}=\frac{1}{2}$

Answer 1

The Correct Option is D

Let x be resistance per unit length of the wire. Then, The resistance of the upper portion is $\, \, \, \, R_1 =xl_1$ The resistance of the lower portion is $\, \, \, \, R_2 =xl_2$ Equivalent resistance between A and B is $\, \, \, \, R=\frac{R_1R_2}{R_1+R_2}=\frac{(xl_1)(xl_2)}{xl_1+xl_2}$ $\, \, \, \, \frac{8}{3} =\frac{xl_1l_2}{l_1+l_2}\,$ or $\, \frac{8}{3}=\frac{xl_1l_2}{l_2\bigg(\frac{l_1}{l_2}+1\bigg)}$ or $\frac{8}{3} =\frac{xl_1}{\bigg(\frac{l_1}{l_2}+1\bigg)}$ ...(i) Also $R_0=xl_1+xl_2$ $\, \, \, \, \, \, 12 = x (l_1+l_2)$ $\, \, \, \, \, \, 12 =xl_2\bigg(\frac{l_1}{l_2}+1\bigg)$ ...(ii) Divide (i) by (ii), we get $\, \, \, \, \frac{\frac{8}{3}}{12} =\frac{\frac{xl_1}{\bigg(\frac{l_1}{l_2}+1\bigg)}}{xl_2\bigg(\frac{l_1}{l_2}+1\bigg)}\,$ or $\, \frac{8}{36}=\frac{l_1}{l_2\bigg(\frac{l_1}{l_2}+1\bigg)^2}$ $ \bigg(\frac{l_1}{l_2}+1\bigg)^2{}\, \frac{8}{36}= \frac{l_1}{l_2}\,$ or $\, \bigg(\frac{l_1}{l_2}+1\bigg)^2\, \frac{2}{9} =\frac{l_1}{l_2}$ Let $y =\frac{l_1}{l_2}$ $\therefore 2(y+1)^2 = 9y \,$ or $\, 2y^2+2+4y =9y $ or $ 2y^2 - 5y + 2 =0 $ Solving this quadratic equation, we get $ \, \, \, \, \, y= \frac{1}{2}\,$ or $ 2\, \therefore \frac{l_1}{l_2} = \frac{1}{2}$

Answer 2

• Two or more resistors can be connected in series or parallel combination to get the equivalent resistance of the circuit.
• A single resistor that draws the same current as the given combination of the resistors when the same potential difference is applied across its endpoints is called Equivalent resistance.

Resistors in Series

Two or more resistors are said to be connected in series if they are connected one after the other such that the same current flows through all the resistors when the same potential difference is applied.

The formula for resistors in series is given by

R = R₁ + R₂ + R₃ + ……….

Resistors in Parallel

Two or more resistors are said to be connected in series if one end of a resistor is connected to one end of another resistor and the second end of the first resistor is connected to the second of the other resistor such that the potential difference across each resistor is equal to the applied potential difference across the combination.

The formula for resistors in parallel is given by

1/R = 1/R₁ + 1/R₂ + 1/R₃ +...........