The coefficient of linear expansion of glass is α8 per ^∘ C and the cubical expansion of mercur

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The coefficient of linear expansion of glass is α8 per ^∘ C...

The coefficient of linear expansion of glass is $\alpha_{8}$ per ${ }^{\circ} \mathrm{C}$ and the cubical expansion of mercury is $\gamma_{m}$ per ${ }^{\circ} \mathrm{C}$. The volume of the bulb of a mercury thermometer at $0^{\circ} \mathrm{C}$ is $V_{0}$ and cross-section of the capillary is $A_{0}$. What is the. length of mercury column in capillary at $T^{\circ} \mathrm{C}$, if the mercury just fills the bulb at $0^{\circ} \mathrm{C}$ ? (a) $\frac{V_{0} T\left(\gamma_{m}+3 \alpha_{p}\right)}{A_{0}\left(1+2 \alpha_{g} T\right)}$ (b) $\frac{V_{0} T\left(\gamma_{m}-3 \alpha_{Q}\right)}{A_{0}\left(1+2 \alpha_{g} T\right)}$ (c) $\frac{V_{0} T\left(\gamma_{m}+2 \alpha_{g}\right)}{A_{0}\left(1+3 \alpha_{g} T\right)}$ (d) $\frac{V_{0} T\left(\gamma_{m}-2 \alpha_{g}\right)}{A_{0}\left(1+3 \alpha_{\varphi} T\right)}$

The coefficient of linear expansion of glass is $\alpha_{8}$ per ${ }^{\circ} \mathrm{C}$ and the cubical expansion of mercury is $\gamma_{m}$ per ${ }^{\circ} \mathrm{C}$. The volume of the bulb of a mercury thermometer at $0^{\circ} \mathrm{C}$ is $V_{0}$ and cross-section of the capillary is $A_{0}$. What is the. length of mercury column in capillary at $T^{\circ} \mathrm{C}$, if the mercury just fills the bulb at $0^{\circ} \mathrm{C}$ ?
(a) $\frac{V_{0} T\left(\gamma_{m}+3 \alpha_{p}\right)}{A_{0}\left(1+2 \alpha_{g} T\right)}$
(b) $\frac{V_{0} T\left(\gamma_{m}-3 \alpha_{Q}\right)}{A_{0}\left(1+2 \alpha_{g} T\right)}$
(c) $\frac{V_{0} T\left(\gamma_{m}+2 \alpha_{g}\right)}{A_{0}\left(1+3 \alpha_{g} T\right)}$
(d) $\frac{V_{0} T\left(\gamma_{m}-2 \alpha_{g}\right)}{A_{0}\left(1+3 \alpha_{\varphi} T\right)}$

Key Concepts

Thermal Expansion

Thermal expansion is the phenomenon where materials change their dimensions (length, area, or volume) as a result of a change in temperature. This effect is crucial in many applications, including thermometers, where both the container and the fluid inside respond to temperature variations.

Coefficient of Linear Expansion

The coefficient of linear expansion quantifies the fractional change in length per unit temperature change for a solid material. It is used to understand how materials like glass change in dimensions, and it plays a key role when considering the expansion of the capillary or bulb walls in devices like mercury thermometers.

Coefficient of Cubical Expansion

The coefficient of cubical (or volumetric) expansion describes the fractional change in volume per degree change in temperature, typically used for fluids. For mercury in a thermometer, this coefficient is critical in determining how the volume of the liquid increases with temperature.

Interplay of Container and Fluid Expansion

In practical devices like thermometers, the expansion of both the container (made of a solid like glass) and its contents (such as mercury) must be accounted for. The container’s linear expansion affects its cross-sectional area, while the fluid's cubical expansion affects its volume. The combined effect determines the resultant change in the length of the fluid column, which is the measurable output of the thermometer.

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