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Question: The resistivity of the material of potentiometer wire is \[5 \times {10^{ - 6}}\Omega m\;\]and its a...

The resistivity of the material of potentiometer wire is 5×10−6Ωm  5 \times {10^{ - 6}}\Omega m\;and its area of cross-section is5×10−6m25 \times {10^{ - 6}}{m^2}. If 0.2  A  0.2\;A\; current is flowing through the wire, then the potential drop per metre length of the wire is
A. 0.1Vm−10.1V{m^{ - 1}}
B. 0.5Vm−10.5V{m^{ - 1}}
C. 0.25Vm−10.25V{m^{ - 1}}
D. 0.2Vm−10.2V{m^{ - 1}}
E. 0.001Vm−10.001V{m^{ - 1}}

Explanation

Solution

We use the relation between resistance and resistivity to find the resistance of the material. Then we use the ohm's law to find the voltage. Dividing the voltage with the length of wire, voltage drop per meter length of wire can be found.

Formula used:
ρ=RAl\rho = \dfrac{{RA}}{l}
Ohm's law:V=iRV = iR
Here,
Resistance is represented by RR
Resistivity is represented by ρ\rho
Length of wire is represented by ll
Area of cross-section is represented by AA
Voltage is represented by VV
Current is represented by ii

Complete step by step answer:
Resistivity is the electrical resistance of a material of unit area of unit length. Resistivity is a constant value for a given material.
Resistivity is equal to
ρ=RAl\rho = \dfrac{{RA}}{l}
From which we can find resistance
R=ρlAR = \dfrac{{\rho l}}{A}
Given, length of the potentiometer, l=1ml=1m, ρ=5×10−6Ωm\rho = 5 \times {10^{ - 6}}\Omega m and A=5×10−6m2A = 5 \times {10^{ - 6}}{m^2}. Putting these values in the formula, we find the resistance.
R=5×10−6×15×10−6R = \dfrac{{5 \times {{10}^{ - 6}} \times 1}}{{5 \times {{10}^{ - 6}}}}
Given, i=0.2Ai=0.2A
From ohm's law
V=iRV = iR
  ⟹  V=0.2×5×10−6×15×10−6\implies V = \dfrac{{0.2 \times 5 \times {{10}^{ - 6}} \times 1}}{{5 \times {{10}^{ - 6}}}}
  ⟹  V=0.2V\implies V = 0.2V
To find the voltage drop per meter we simply divide the voltage by the length
V=0.21Vm−1=0.2Vm−1V = \dfrac{{0.2}}{1}V{m^{ - 1}} = 0.2V{m^{ - 1}}
Hence the voltage drop per meter of wire is 0.2Vm−10.2V{m^{ - 1}}.

So, the correct answer is “Option D”.

Note:
The potential drop for one meter or material is also called the potential gradient. It is the potential difference between two points on the wire that are one meter apart. Hence by dividing voltage by the length of wire we find the potential gradient. The units are volts per meter.
Resistivity depends on the type of material. The resistivity of two wires of the same material is always constant. Resistivity does not depend on the mass or length of the substance; it is a fixed value for different materials (like specific heat capacity) hence it is an intrinsic property. It purely depends on the nature of the element.