Question

a uniform wire of resistance 20 ohm having resistance 1 ohm/m is bent in form of a circle as shown in the figure. If the equailent resistance between M and N is 1.8 ohm then find the length of the shorter section.​

Answer 1

Concept Introduction: Resistance is given by Potential difference by per unit current.

Given:

We have been Given: Resistance of wire

$20 \: ohm$

Resistance through the wire

$1 \: ohm {m}^{ - 1}$

Equivalent Resistance through Point M and N is

$1.8 \: ohm$

To Find:

We have to Find: Length of the section of wire from M to N.

Solution:

According to the problem, Total Length of the wire is

$l = r \div rho$

that will give,

$l = 20 \: ohm \div 1 \: ohm {m}^{ - 1} = 20 \: m$

Let, the length of the short section be

$x \: m$

therefore the remaining length is

$(20 - x) \: m$

Therefore the resistance of the lengths will be

$r1 = x \: ohm$

$r2 = (20 - x) \: ohm$

Given, the figure the resistances are in parallel, so,

$r(equivalent) = \frac{r1 \times r2}{r1 + r2}$

therefore putting the values,

$1.8 = \frac{x(20 - x)}{ x + 20 - x} = \frac{20x - {x}^{2} }{20}$

cross multiplying,

$20 \times 1.8 = 20x - {x}^{2}$

therefore,

${x}^{2} - 20x + 36 = 0 \\ {x}^{2} - 18x - 2x + 36 = 0 \\ x(x - 18) - 2(x - 18) = 0 \\ (x - 2)( x - 18) = 0$

therefore,

$x = 2 \: or \: 18$

therefore length of the shorter sider is,

$2 \: m$

Final Answer: The Length of the Shorter Side is

$2 \: m$

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