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Energy stored in a stretched wire

Energy stored

​When a body is deformed, work is done by the deforming forces against the intermolecular force of attraction between the molecules of the body. This work done is stored in the body in the form of elastic potential energy.
​Consider a body of length L and of cross section area A. Suppose on applying force F, elongation x is produced in the body.
​Then,  the young's modulus of elasticity  of the material of the body is given by,
Y=FLAx
F=YAL


Then, small work done to produce an infinitesimally small extension dx is given by,
dW=Fdx
Integrating within limits from 0 to x,
W=0xFdx
W=0xYALdx
W=YAL0xxdx
W=YAL[x22]0x
W=12YALx²
W=12YALxx
W=12Fx
W=12×force×extension
Since, work done = energy stored
Energy stored=12×force×extension

Energy Density

Energy density is defined as the elastic potential energy per unit volume of the deformed body.
​Thus, we have,
Energy Density (U)=12FxV
here, V = Volume of the deformed body and
V=Area of cross section×length
V=A×L
Energy Density (U)=12FxAL
Here,
FA=normal stress

and,
xL=Longitudinal strain