Understanding Thermal Expansion and Constant Length Difference

The question asks about the relationship between the initial lengths of a copper rod and a steel rod and their coefficients of linear expansion, such that the difference in their lengths remains constant at all ambient temperatures.

When a material is heated, its length increases due to thermal expansion. The change in length ($\Delta L$) of a rod of original length ($L_0$) is given by the formula:

\(\Delta L = L_0 \alpha \Delta T\)

where:

• \(L_0\) is the original length at a reference temperature.
• \(\alpha\) is the coefficient of linear expansion of the material.
• \(\Delta T\) is the change in temperature.

The new length ($L$) at temperature \(T_0 + \Delta T\) is given by:

\(L = L_0 + \Delta L = L_0 (1 + \alpha \Delta T)\)

Setting up the Condition for Constant Length Difference

Let the initial lengths of the copper rod and the steel rod at some reference temperature be \(L_C\) and \(L_S\) respectively. Let their coefficients of linear expansion be \(\alpha_C\) and \(\alpha_S\). When the temperature changes by \(\Delta T\), the new lengths will be:

• New length of copper rod: \(L_C' = L_C (1 + \alpha_C \Delta T)\)
• New length of steel rod: \(L_S' = L_S (1 + \alpha_S \Delta T)\)

The condition given is that the difference between their lengths is the same at all ambient temperatures. This means the difference in the new lengths must be equal to the difference in the original lengths:

\(L_C' - L_S' = L_C - L_S\)

Substitute the expressions for \(L_C'\) and \(L_S'\):

\(L_C (1 + \alpha_C \Delta T) - L_S (1 + \alpha_S \Delta T) = L_C - L_S\)

Deriving the Relationship between Lengths and Coefficients

Expand the equation:

\(L_C + L_C \alpha_C \Delta T - L_S - L_S \alpha_S \Delta T = L_C - L_S\)

Subtract \(L_C - L_S\) from both sides of the equation:

\(L_C \alpha_C \Delta T - L_S \alpha_S \Delta T = 0\)

Factor out \(\Delta T\):

\((L_C \alpha_C - L_S \alpha_S) \Delta T = 0\)

This equation must hold true for any arbitrary change in temperature \(\Delta T\) (except \(\Delta T = 0\)). Therefore, the term in the parenthesis must be zero:

\(L_C \alpha_C - L_S \alpha_S = 0\)

Rearrange the terms to find the relationship between \(L_C\), \(L_S\), \(\alpha_C\), and \(\alpha_S\):

\(L_C \alpha_C = L_S \alpha_S\)

Now, we can express the ratio of the initial lengths:

\(\frac{L_C}{L_S} = \frac{\alpha_S}{\alpha_C}\)

Comparing with Options

Let's compare our derived relationship with the given options:

• Option 1: \(\rm \frac{L_C}{L_S}=\frac{\alpha_S}{\alpha_C}\). This matches our result.
• Option 2: \(\rm L_C - L_S = \alpha_C - \alpha_S\). This is not the correct relationship.
• Option 3: \(\rm \frac{L_C}{L_S}=(\frac{\alpha_C}{\alpha_S})^{1/2}\). This is not the correct relationship.
• Option 4: \(\rm \frac{L_C}{L_S}=\frac{\alpha_C}{\alpha_S}\). This is the inverse of our required ratio.

Therefore, for the difference between the lengths of the copper rod and the steel rod to remain constant at all ambient temperatures, the ratio of their initial lengths must be equal to the inverse ratio of their coefficients of linear expansion.

The correct relationship is \(\frac{L_C}{L_S} = \frac{\alpha_S}{\alpha_C}\).