Problem 84 - Master Resource Book in JEE Main Physics

Question

Four identical rods of same material are joined end to end to form a square. If the temperature difference between the ends of a diagonal is \(100^{\circ} \mathrm{C}\) then the temperature difference between the ends of other diagonal will be (a) \(0^{\circ} \mathrm{C}\) (b) \(\frac{100}{l}{\underline{\phantom{xx}}}^{\circ} \mathrm{C} ;\) where \(l\) is the length of each rod (c) \(\frac{100}{2 l}^{*} \mathrm{C}\) (d) \(100^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

0°C; symmetry ensures no temperature gradient across the other diagonal.

Step 1: Understanding the Problem

We have a square formed by four identical rods of the same material. The temperature difference across one diagonal is given as \( 100^{\circ} \mathrm{C} \). We need to find the temperature difference across the other diagonal.

Step 2: Geometry of the Square

When four rods are joined to form a square, the diagonals of the square intersect at right angles and divide each diagonal into two equal parts.

Step 3: Symmetry of the Square

Due to the symmetrical nature of the square and identical material properties of the rods, the heat distribution will be symmetrical. The ends of the other diagonal will be at the same average temperature.

Step 4: Analyzing Temperature Distribution

The temperature difference across one diagonal being \( 100^{\circ} \mathrm{C} \) implies that there is a temperature gradient along this diagonal. Due to symmetry, the midpoint of the diagonal has uniform temperature distribution from all sides.

Step 5: Conclusion from Symmetry

The temperature difference across the other diagonal is \( 0^{\circ} \mathrm{C} \) because the heat distribution allows for an equal temperature throughout, with no gradient forming across this diagonal.

Key Concepts

Heat Distribution

In the context of heat distribution, consider a square formed by identical rods. When one diagonal of this square shows a temperature difference, analyzing the heat distribution becomes crucial. Key to understanding this is how heat flows along the rods. - Heat tends to move from higher to lower temperatures, creating a temperature gradient. - For the rods forming a square, heat is distributed symmetrically, radiating from hotter to cooler areas along the diagonal. If we know one diagonal holds a 100°C difference, this means a heat flow occurs across it. However, due to the symmetrical nature of the square setup, this doesn't imply all diagonals experience changes in temperature the same way. The rods' material properties, constant throughout, support uniform heat transfer. Thus, the heat distribution is governed by the symmetry and uniform thermal properties of the rods, ensuring the temperature change is primarily restricted to the given diagonal.

Symmetrical Properties

Understanding the symmetrical properties of a square helps explain why temperature differences occur across its diagonals differently. The square shape inherently supports symmetry: - Each rod is the same, allowing uniform heat conduction. - Diagonals split the square equally, each forming equal angles where they intersect. This symmetry is not just geometrical but thermal as well. The identical nature of the rods ensures that any heat applied distributes uniformly within the shape's constraints. Because of these symmetrical properties, while the diagonal with the initial temperature difference experiences a gradient, the other diagonal is unaffected. The symmetrical setup means the heat has already balanced itself in the rest of the structure. There’s no new temperature change along the other diagonal, leading to a 0°C difference, according to symmetry principles.

Thermal Conductivity

Thermal conductivity is a crucial factor that determines how temperature changes are distributed in a material. In this context, consider each rod made of the same material: - The uniform thermal conductivity allows for even heat flow. - Atoms within the rods aid in energy transfer, pushing heat uniformly along the lengths. As a result, any temperature gradient introduced reflects uniformly, especially on a symmetrical structure like a square. The fact that the rods are of equal length and material ensures that heat is spread evenly. Thus, when examining the square's alternate diagonal, the overall thermal conductivity causes any effects from the temperature change to neutralize each other. Consequently, no temperature difference appears across this diagonal, because the material aids in achieving a state of thermal equilibrium quickly and efficiently.