Question

Two wires A and B have equal lengths and are made of the same material, but the diameter of A is twice that of B. Then, for a given load
(A). The extension of A will be four times that of B
(B). The extensions of A and B will be equal
(C). The strain in A will be half that of B
(D). The strains in A and B will be equal

Answer 1

Hint: For determining the value of extension in both wires, we will use the concept of Young’s Modulus. As the given wires have different diameters, we will find out the relation between the extensions in two wires.

Formula used:
$Y=\dfrac{F}{\dfrac{Al}{\Delta l}}=\dfrac{F\Delta l}{Al}$
$\text{Strain = }\dfrac{\Delta l}{l}$

Complete step by step answer:
Given two wires A and B having equal lengths but diameter of A is twice that of B. For finding the relation in extension in the wires, we will use the equation of Young’s Modulus. Young’s Modulus, being an internal property of an object, does not depend upon the shape and size of the body. As both the wires are made of the same material, both will have equal value of Young’s Modulus.
$Y=\dfrac{F}{\dfrac{Al}{\Delta l}}=\dfrac{F\Delta l}{Al}$
Let’s take Young’s Modulus of wire A and wire B as ${{Y}_{A}}$ and ${{Y}_{B}}$ respectively.
Diameter of wire A is twice that of B, so area of cross section of A will be 4 times that of B
If cross sectional area of B is $A$, then that of A is $4A$.
${{Y}_{A}}=\dfrac{F\Delta {{l}_{A}}}{4Al}$
${{Y}_{B}}=\dfrac{F\Delta {{l}_{B}}}{Al}$
Comparing equations of Young’s Modulus for wire A and wire B
${{Y}_{A}}={{Y}_{B}}$
We get,
$\begin{aligned} & \dfrac{F\Delta {{l}_{A}}}{4Al}=\dfrac{F\Delta {{l}_{B}}}{Al} \\\ & \Delta {{l}_{A}}=4\Delta {{l}_{B}} \\\ \end{aligned}$
Extension in wire A will be 4 times the extension in wire B.
Strain is the ratio of the extension in wire to the original length of wire or we can say the ratio of change in length to the original length.
$\text{Strain = }\dfrac{\Delta l}{l}$
As we have calculated above, $\Delta {{l}_{A}}=4\Delta {{l}_{B}}$
Therefore, strain in wire A is also 4 times the strain in wire B.
Hence, the correct option is A.

Note: Students should keep in mind that Young’s modulus does not depend on shape and size of body; it is an internal property of a material. For wire A and B, the value of Young’s modulus will be the same.