Question

# Two wires $$A$$ and $$B$$ have equal lengths and are made of the same material, but the diameter of wire $$A$$ is twice that of wire $$B$$. Then, for a given load:

A
the extension of B will be four times that of A
B
the extensions of A and B will be equal
C
the strain in B is four times that in A
D
the strains in A and B will be equal

Solution

## The correct options are A the extension of $$B$$ will be four times that of $$A$$ C the strain in $$B$$ is four times that in $$A$$$$Y=\dfrac{F/\pi \left ( \dfrac{d}{2} \right )^2}{\Delta l/l}$$$$Y=\dfrac{F/\pi \left ( \dfrac{d_A}{2} \right )^2}{\Delta l_A/l}$$-----(1)$$Y=\dfrac{F/\pi \left ( \dfrac{d_B}{2} \right )^2}{\Delta l_B/l}$$-----(2)Dividing (1) by (2)$$1=\dfrac{\pi \left ( \dfrac{d_B}{2} \right )^2}{\pi \left ( \dfrac{d_A}{2} \right )^2}\times \dfrac{\Delta l_B}{\Delta l_A}$$$$1=\left ( \dfrac{d_B}{d_A} \right )^2\times \dfrac{\Delta l_B}{\Delta l_A}$$$$1=\left ( \dfrac{1}{2} \right )^2\times \dfrac{\Delta l_B}{\Delta l_A}$$$$\dfrac{\Delta l_A}{\Delta l_B}=\dfrac{1}{4}$$strain$$=\dfrac{\Delta l}{l}$$So, extension in B will be four times of A and strain in B will be four times of A since $$\Delta_B=4\Delta_A$$ and length of A and B are same.Physics

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