# For anisotropic solids, coefficient of volume expansion is most appropriately given as (where symbols have their usual meanings)

A. \[3{\alpha _x}\]

B. \[{\alpha _x} + {\alpha _y} + {\alpha _z}\]

C. \[2{\alpha _x} + 2{\alpha _y}\]

D. \[3{\alpha _z} + {\alpha _x} - {\alpha _y}\]

Answer

Verified

279.9k+ views

**Hint:**Find the coefficient of linear expansion for each sides and then use the formula to find the coefficient of volume expansion then compare them to get the relation between the coefficient of volume expansion to linear expansion.

**Formula used:**

The linear expansion of any solid is given by,

\[l' = {l_0}(1 + \alpha \Delta T)\]

where, \[l'\] is the expanded length of the solid during change in temperature \[\Delta T\] , \[{l_0}\] is the initial length and \[\alpha \] is the expansion coefficient.

The volume expansion of any solid,

\[V' = {V_0}(1 + \gamma \Delta T)\]

where, \[V'\]is the expanded volume of the solid during change in temperature \[\Delta T\] , \[{V_0}\] is the initial length and \[\gamma \] is the expansion coefficient.

**Complete step by step answer:**

We have here an anisotropic solid and we have to find the relation between the coefficient of volume expansion and the coefficient of linear expansion. Now, since anisotropic solids stretch in length in different directions.

So, let the change in length along different axes are,

Change along the x axis: \[{l_x}' = {l_{x0}}(1 + {\alpha _x}\Delta T)\]

Change along the y axis: \[{l_y}' = {l_{y0}}(1 + {\alpha _y}\Delta T)\]

Change along the z axis: \[{l_z}' = {l_{z0}}(1 + {\alpha _z}\Delta T)\]

So, multiplying this three equations we will have,

\[{l_x}'{l_y}'{l_z}' = {l_{z0}}(1 + {\alpha _z}\Delta T){l_{y0}}(1 + {\alpha _y}\Delta T){l_{x0}}(1 + {\alpha _x}\Delta T)\]

Or, \[{l_x}'{l_y}'{l_z}' = {l_{z0}}{l_{y0}}{l_{x0}}(1 + {\alpha _z}\Delta T)(1 + {\alpha _y}\Delta T)(1 + {\alpha _x}\Delta T)\]

Now, neglecting the higher order terms of \[\alpha \] (since the dimension of \[\alpha \]is very small so, \[{\alpha ^2},{\alpha ^3} < < \alpha \])

We will have,

\[{l_x}'{l_y}'{l_z}' = {l_{z0}}{l_{y0}}{l_{x0}}[1 + ({\alpha _x} + {\alpha _y} + {\alpha _z})\Delta T]\]

Now, the \[{l_x}'{l_y}'{l_z}'\] is equal to the volume after the change and \[{l_{z0}}{l_{y0}}{l_{x0}}\] is the initial volume.

So, replacing them we will have,

\[V' = {V_0}[1 + ({\alpha _x} + {\alpha _y} + {\alpha _z})\Delta T]\]

Now, we know that the volume expansion of any solid,

\[V' = {V_0}(1 + \gamma \Delta T)\]

where, \[V'\] is the expanded volume of the solid during change in temperature \[\Delta T\] , \[{V_0}\] is the initial length and \[\gamma \] is the volume expansion coefficient.

So, comparing these two equations we can write,

\[\therefore \gamma = ({\alpha _x} + {\alpha _y} + {\alpha _z})\]

Hence, the coefficient of volume expansion of anisotropic solid to linear expansion of is given by \[({\alpha _x} + {\alpha _y} + {\alpha _z})\]

**Hence, option B is the correct answer.**

**Note:**The relation of coefficient of volume expansion and linear expansion for different types of solid is different. Here, we can see that for anisotropic solid it is different in different direction but for isotropic solid it is same in every direction for that we will have the relation as, \[\gamma = 3{\alpha _x} = 3{\alpha _y} = 3{\alpha _z}\].

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Define absolute refractive index of a medium

Which of the following would not be a valid reason class 11 biology CBSE

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What is meant by monosporic development of female class 11 biology CBSE

Trending doubts

The ray passing through the of the lens is not deviated class 10 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

What is the nlx method How is it useful class 11 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the difference between anaerobic aerobic respiration class 10 biology CBSE