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Resistance of a conductor of length $75\,cm$ is $3.25\Omega $. What will be the length of a similar conductor whose resistance is $13.25\Omega $ $?$
(A) $305.76\,cm$
(B) $503.76\,cm$
(C) $200\,cm$
(D) $610\,cm$

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Last updated date: 13th Jun 2024
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Answer
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Hint We know the value of conductor length and resistance of any conductive material here, so by using this formula we calculate the end-to-end resistance of a conductor given length and area, we also know that specific resistance is given in units of ohms for materials.
Useful formula
Resistance of conductor is given below,
$R = \rho * \left( {\dfrac{L}{A}} \right)$
$R = $ resistance in ohms
$\rho = $ material resistivity in ohms per meter
$L = $ conductor length in meters
$A = $ cross-sectional area in square meters

Complete step by step procedure
Given by,
Resistance of a conductor of length $75\,cm$is \[3.25{\text{ }}ohm\]
\[R = 3.25{\text{ }}ohm\] and \[I = 75\,cm\]
Let length of another conductor whose resistance is \[16.25{\text{ }}ohm\]be $x$
\[R{\text{ }} = {\text{ }}16.25{\text{ }}ohm\] \[I = \,x\,cm\]
A conductor's resistance and the conductor's length are directly proportional to one another. The resistance of the conductor also increases as the length of the conductor increases.
The relationship between a conductor's length and resistance is given as $R = \rho * \left( {\dfrac{L}{A}} \right)$
The material of the conductor remains the same, so there is constant resistivity $(\rho )$.
According to that,
The given value in substituting the equation.
So,
$3.25 = \rho \dfrac{{75}}{A}$……………………………. $(1)$
For the resistance to be $13.25$ ohms.
The equation is written as,
$13.25 = \rho \dfrac{x}{A}$……………………………$(2)$
Solving the both equations,
We get,
\[x = 305.76\,cm\]
Hence,
The length of the conductor for it have $13.25\Omega $ resistance is $305.76\,cm$

Thus, option A is the correct answer.

Note As electrons pass through a conductor, such as a metal wire, an electrical current flow. With the ions in the metal, the moving electrons will collide. It makes it harder for the current to flow. All materials have some resilience, but some materials are more or less resistant to electric current flow than other materials.