Question

When a piece of aluminum wire of finite length is drawn through a series of dies to reduce its diameter to half its original value, its resistance will become:-
A) Two times
B) Four times
C) Eight times
D) Sixteen times

Answer 1

Hint: Resistance of a wire is directly proportional to its length and inversely related to its area. While drawing a wire, its volume and resistivity remain constant. This principle can be used here. Convert the resistance equation in terms of volume. The area of a wire is proportional to the square of its diameter. When the diameter is reduced to half, the area of the wire changes, thereby its resistance also changes.
Formula used:
\[V=Al\]
\[R=\rho \dfrac{l}{A}\]
\[A=\pi {{r}^{2}}=\pi {{\left( \dfrac{d}{2} \right)}^{2}}\]

Complete step-by-step solution:
Electrical resistance R of a wire is directly proportional to its length \[l\] and inversely proportional to its cross-sectional area \[A\]. Also, it depends on the material with which the wire is made (specific electrical resistance or resistivity \[\rho \]).
Resistance,\[R=\rho \dfrac{l}{A}\] -------- 1
While stretching a conductor, its resistivity and the total volume (V) remains constant.
Convert equation 1 in terms of volume, by multiplying numerator and denominator with area A. Then,
\[\Rightarrow R=\dfrac{\rho l\times A}{A\times A}=\dfrac{\rho V}{{{A}^{2}}}\]
(\[V=Al\])
From the above equation, we can see that,
\[ \Rightarrow R\propto \dfrac{1}{{{A}^{2}}}\]
We have,
\[A=\pi {{r}^{2}}=\pi {{\left( \dfrac{d}{2} \right)}^{2}}\]
\[\Rightarrow A\propto {{d}^{2}}\]
Then,
\[\Rightarrow R\propto \dfrac{1}{{{d}^{4}}}\]
 Hence, if we reduce the diameter to half,
\[\Rightarrow R'\propto \dfrac{1}{{{\left( \dfrac{d}{2} \right)}^{4}}}\]
\[R\] increases to \[16R\].
The answer is option D

Note: The resistance of an object depends on its shape and the material with which it is composed. The resistivity of a material is dependent upon the electronic structure of the material and its temperature. For most of the materials, resistivity increases with increasing temperature. But, in the case of conductors, the resistivity increases with increasing temperature. Since the atoms vibrate more rapidly over larger distances at higher temperatures, the moving electrons through a metal make more collisions, and its resistivity increases.