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A wire of cross-sectional area $5.0 \times {10^{ - 6}}{m^2}$ is made of a metal of resistivity $50 \times {10^{ - 8}}\Omega m$. The potential difference across the wire is $6.0V$ and the current is $3.0A$. What is the length of the wire?
A. $0.050m$
B. $0.20m$
C. $5.0m$
D. $20m$

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Last updated date: 13th Jun 2024
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Answer
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Hint: Resistivity is an inherent property of all the materials which determines whether the material is a conductor or an insulator. It is defined as the resistance offered by a conductor of $1m$ length and a cross-sectional area of $1{m^2}$.

Complete step by step answer:
Electrical resistance is a property of any material by which that material tries to resist the flow of electric current. It basically depends upon an inherent property of any material known as Resistivity. The resistivity of any material tells whether the material is a conductor or an insulator. If the resistivity of a material is very high, the material is an insulator. But if its resistivity is low, the material is a conductor. Its SI unit is ohm-meter$\left( {\Omega m} \right)$. Electrical resistance can be calculated with the help of resistivity by,
$R = \rho \dfrac{l}{A}$ . . . . . . . . [Equation 1]
Where, $R = $Electrical resistance of the conductor, $\rho = $Resistivity of the conductor,$l = $Length of the conductor and $A = $Cross-sectional Area of the conductor.

Now, according to Ohm’s Law, the potential difference across any conductor is directly proportional to the electrical current flowing through the conductor. That is,
$V \propto i$
$ \Rightarrow V = iR$
$ \Rightarrow R = \dfrac{V}{i}$ . . . . . . . . [Equation 2]
Where, $V = $Potential difference across a conductor and $i = $Electric current flowing through a conductor.
Equating both the equations,
$\dfrac{V}{i} = \rho \dfrac{l}{A}$
Putting all the known values,
$\dfrac{6}{3} = 50 \times {10^{ - 8}} \times \dfrac{l}{{5 \times {{10}^{ - 6}}}}$
$\therefore l = 20m$

Therefore the correct answer is option D.

Note:When the resistivity of a material is neither extremely high nor extremely low, but is of an intermediary value, then that material is known as a semiconductor. It does not conduct electricity in ideal conditions but when a little energy is provided to it, it starts conducting electricity.