Dumet wire was developed to allow for the expansion of glass in lightbulbs. The wire consists of a core of nickel – steel (invar ) surrounded by a sheath of copper. The diameter of the core and of the sheath are chosen so that the wire duplicates the expansion characteristics of glass. (a) show that the ratio of the nickel –steel radius to that of the copper sheath should be.

To understand the relationship between the nickel-steel core and the copper sheath in Dumet wire, we need to delve into the concept of thermal expansion. The goal here is to ensure that both materials expand at similar rates when heated, which is crucial for maintaining the integrity of lightbulbs. Let's break this down step by step.

Understanding Thermal Expansion

When materials are heated, they tend to expand. This expansion can be quantified using the coefficient of linear expansion, which measures how much a material expands per degree of temperature change. For glass, the coefficient of linear expansion is relatively low compared to metals like copper and nickel-steel.

Materials Involved

• Nickel-Steel (Invar): This alloy has a very low coefficient of thermal expansion, making it suitable for applications where dimensional stability is critical.
• Copper: Copper has a higher coefficient of thermal expansion, which means it expands more than nickel-steel when heated.

Establishing the Ratio

To ensure that the Dumet wire behaves similarly to glass when subjected to temperature changes, we need to establish a ratio between the radius of the nickel-steel core and the copper sheath. Let’s denote:

• r_core: Radius of the nickel-steel core
• r_sheath: Radius of the copper sheath

The ratio we are looking for can be expressed as:

Ratio = r_core / r_sheath

Calculating the Ratio

To find this ratio, we can use the relationship between the coefficients of linear expansion of the materials involved. The condition for the wire to expand similarly to glass can be expressed mathematically as:

α_glass = (α_copper * r_sheath + α_invar * r_core) / (r_core + r_sheath)

Where:

• α_glass is the coefficient of linear expansion of glass.
• α_copper is the coefficient of linear expansion of copper.
• α_invar is the coefficient of linear expansion of nickel-steel.

By rearranging this equation, we can derive the ratio of the radii:

r_core / r_sheath = (α_glass - α_copper) / (α_invar - α_glass)

Example Calculation

Let’s assume the following approximate values for the coefficients of linear expansion:

• α_glass ≈ 9 x 10^-6 °C^-1
• α_copper ≈ 16 x 10^-6 °C^-1
• α_invar ≈ 1 x 10^-6 °C^-1

Plugging these values into our ratio formula:

r_core / r_sheath = (9 x 10^-6 - 16 x 10^-6) / (1 x 10^-6 - 9 x 10^-6)

This simplifies to:

r_core / r_sheath = (-7 x 10^-6) / (-8 x 10^-6) = 0.875

Final Thoughts

This ratio indicates that the radius of the nickel-steel core should be approximately 87.5% of the radius of the copper sheath to ensure that the Dumet wire expands in harmony with glass. This careful design allows for the reliable performance of lightbulbs, preventing issues like breakage or loss of vacuum due to differential expansion.