Question

Isometric view of a solid copper object is shown below. If a constant heat source of 100 degrees C is applied at the point P continuously, which point on the solid will reach the temperature of the heat source the earliest? Neglect heat losses and assume point P lies on an equilateral triangle.

• A. A
• B. B
• C. C
• D. D

Hint:

Imagine a thermal front spreading through the uniform copper from P. Compare the shortest straight-line distance through the solid from P to each labelled point.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question belongs to the topic of

Thermal Physics and Heat Conduction.
We are given a solid copper object with a continuous 100-degree heat source applied at a corner point P. We must determine which of the four points (A, B, C, D) will reach this temperature first, assuming no heat is lost to the environment.

Step 2: Key Formula or Approach:
The rate of heat conduction through a solid medium is governed by Fourier's Law of Heat Conduction:
\[ \frac{dQ}{dt} = -k A \frac{dT}{dx} \]

• The time taken for heat to diffuse and raise the temperature at a distant point is directly proportional to the square of the distance \(x^2\) from the heat source.

• Therefore, the point that is geographically closest to the heat source P through the solid copper body will reach the target temperature earliest.


Step 3: Detailed Explanation:

• Let us analyze the spatial distances from the heat source P to each point:
- P is located at the bottom-right corner of the block.
- Point D is located on the top-right-back edge of the block. The straight-line distance through the copper from P to D is the shortest among all options.
- Point C is on the lip of the central groove, which is further to the left.
- Point B is on the flat platform near the back-left, which is even further away.
- Point A is on the far-left ramp, representing the longest thermal path from P.

• Because copper is a highly isotropic thermal conductor, heat spreads spherically outwards from P. The thermal front will reach D first due to its proximity.


Step 4: Final Answer:
Point D will reach the heat source temperature earliest, which corresponds to Option (D).

Answer 2

Concept:

• In one uniform copper solid, thermal diffusivity is the same in every direction.
• A heat disturbance reaches a point sooner when the shortest distance through the solid from the source is smaller, since diffusion time scales as $t\propto L^2$.

Step 1: Use P as the centre of expanding thermal fronts.
Imagine equal-temperature fronts spreading through the copper from P. The first labelled point touched by these fronts will heat earliest.

Step 2: Compare the positions of the labelled points.
A, B, and C lie progressively farther toward the left side of the object. D lies on the nearest upper edge directly above the right-hand region containing P.

Step 3: Compare the conduction distances.
The distance through copper from P to D is shorter than the corresponding distances from P to C, B, or A.
Because $t\propto L^2$, the shortest path gives the earliest temperature rise.

Final Answer: Point D, option D.