Physics THERMAL EXPANSION

Thermal Expansion :

Thermal expansion can be defined as expansion due to increase in temperature. When temperature of a body increases, the distance between any two points on the object increases. The expanded object is like a photographic enlargement. Figure shows a plate with a hole in it. The hole expands in the same proportion as the metal, it does not get smaller.

At atomic level, thermal expansion may be explained by considering how the potential energy of the atoms varies with distance.
The equilibrium position of an atom will be at the minimum of the potential energy well if the well is symmetric. But if the potential energy curve is asymmetric about the `r_0`, then the average position of an atom will not be at the minimum point. When the temperature is raised the amplitude of the vibration increases and the average position is located at a greater interatomic separation. This increased separation is certified as expansion in the material.

Linear Expansion :

When temperature of a rod increases, length of rod also increases. The increase in length is directly proportional to its original length `L_0` and change in temperature `DeltaT`.

`dL=alpha L_0 dT=> Delta L =alphaL_0 Delta T`

if `alpha` is constant & `alpha Delta t < < 1`

`alpha=lim_(Delta T->0) 1/L_0 (Delta L)/(Delta T)`

Where, `alpha` is called the coefficient of linear expansion whose unit is `text()^0C^(-1)` or `K^(-1)` .

`L = L_0(1 +alpha Delta T)` Where, `L` is the length after heating the rod.

`text(Variation of)` `alpha` `text(with distance and temperature)`

a. If `alpha` varies with distance, (say `alpha = ax+ b`)
Choose an element 'dx' at a distance of `x`, expansion in this element 'dx' can be written as

`dl=dxalphaDeltaT`

Total expension `DeltaL=intdl=intalphaDeltaTdx=DeltaTint(ax+b)dx`

b. If `alpha` varies with temperature, (say `alpha =aT + b`)

Then total expansion `DeltaL=intalphaL_0dT=int(aT+b)L_0dT`

Superficial or Areal Expansion :

When a solid plate is heated and its area increases, then the thermal expansion is called superficial or areal expansion. The increase in area is directly proportional to its original area `A_0` and change in temperature `Delta T` .

`dA=beta A_0 dT=> Delta A=beta A_0 Delta T` if `beta` is consatant & `beta Delta T < < 1`

`beta = lim_(Delta T-> 0) 1/A_0 (Delta A)/(Delta T)`

Where, `beta` is called the coefficient of areal expansion whose unit is `text()^0C^(-1)` or `K^(-1)`

`A=A_0(1+beta Delta T)` Where, `A` is the area after heating the plate.

Cubical or Volume Expansion :

When a solid is heated and its volume increases, then the thermal expansion is called volume or cubical expansion. The increase in volume is directly proportional to its original volume `V_0` and change in temperature `Delta T` .

`dV=gamma V_0 dT=> Delta V=gamma V_0 Delta T` if `gamma ` is constant & `gamma Delta T < < 1`

`gamma =lim_(Delta T-> 0)1/V_0 (Delta V)/(Delta T)`

Where `gamma` is called the coefficient of cubical expansion whosse unit is `text()^0 C^(-1)` or `K^(-1)`

`V=V_0(1+ gamma Delta T)` Where, `V` is the volume after heating the solid.

Relation among `,beta` and `gamma`

For isotropic bodies

`alpha:beta:gamma=1:2:3`