Animated Solution for Physics - Properties of Solids and Liquids: Two identical conducting rods are first connected independently to two vessels, one containing water at 100°C and the other containing ice at 0°C. In the second case, the rods are joined end to end and connected to the same vessels. Let $q_1$ and $q_2$ gram per second be the rate of melting of ice in the two cases respectively. The ratio $\frac{q_1}{q_2}$ is

Imagine you are tasked with melting ice as quickly as possible using only the heat from a boiling pot of water. You are given two identical metal rods to transfer the heat. Should you connect them side-by-side (in parallel) or end-to-end (in series)? This classic physics problem beautifully intertwines the concepts of thermal conduction and phase change, revealing a profound analogy with electrical circuits.

Let's embark on a journey to decode the mathematics behind this setup and discover why one configuration is vastly superior to the other.

A Powerful Tool

In physics, we often borrow mathematical frameworks from one domain to solve problems in another. Thermal conduction is strikingly similar to electrical conduction.

Just as electrical current $I=\frac{dq}{dt}$ is driven by a voltage difference $\Delta V$ across an electrical resistance $R$, the heat current $H=\frac{dQ}{dt}$ is driven by a temperature difference $\Delta T$ across a thermal resistance $R_{th}$.

For a uniform rod of length $L$, cross-sectional area $A$, and thermal conductivity $K$, the thermal resistance is defined as $R_{th}=\frac{L}{KA}$. Because our two rods are identical, they both possess the exact same thermal resistance, which we will simply call $R$.

Our goal is to find the rate at which the ice melts, denoted by $q=\frac{dm}{dt}$. But how does the melting rate connect to the heat current?

When heat reaches the ice at $0^{\circ }\text{C}$, it doesn't raise the temperature; instead, it breaks the molecular bonds, causing a phase change from solid to liquid. The heat required for this transformation is governed by the latent heat of fusion, $L_f$:

If we divide both sides by the infinitesimal time interval $dt$, we get a direct relationship between the heat current and the melting rate:

Now, let's substitute our thermal resistance formula into this equation:

Here is the crucial insight: In both configurations (parallel and series), the temperature difference $\Delta T$ is fixed at $100^{\circ }\text{C}-0^{\circ }\text{C}=100^{\circ }\text{C}$. Furthermore, the latent heat $L_f$ is a constant property of water. Because both $\Delta T$ and $L_f$ are constants, we can establish a powerful proportionality:

The rate of melting is strictly inversely proportional to the equivalent thermal resistance of the setup. To find the ratio of the melting rates, we simply need to find the inverse ratio of their thermal resistances!

Parallel Rods

In the first case, the rods are connected "independently" to the two vessels. This means each rod spans the entire temperature gap from $100^{\circ }\text{C}$ to $0^{\circ }\text{C}$. This is the exact definition of a parallel connection.

Just like electrical resistors in parallel, the equivalent thermal resistance $R_{eq1}$ is found using the reciprocal sum:

Flipping the fraction gives us the equivalent resistance for the first case:

By providing two independent paths for the heat to flow, the overall resistance is halved, allowing heat to surge through rapidly.

Series Rods

In the second case, the rods are joined end-to-end. The heat must painstakingly travel through the entire length of the first rod, and then continue through the entire length of the second rod. This is a series connection.

For resistors in series, the equivalent resistance $R_{eq2}$ is simply the direct sum of the individual resistances:

By forcing the heat through a longer, restricted path, the overall resistance is doubled, severely choking the heat flow.

Calculating the Ratio

We are now armed with everything we need to find the ratio of the melting rates, $\frac{q_1}{q_2}$. Using our master proportionality $q\propto \frac{1}{R_{eq}}$, we can write:

Substitute the equivalent resistances we just calculated:

The $R$ terms elegantly cancel out, and the denominator's $2$ flips up to multiply the numerator:

This result is profound. By simply rearranging the same two rods from series to parallel, you don't just double the melting rate—you quadruple it! This happens because the parallel setup halves the resistance, while the series setup doubles it, creating a factor of four difference between the two extremes.

Physics is not just about plugging numbers into formulas; it's about recognizing the underlying symmetries of nature, like the beautiful mirror between electrical circuits and the flow of heat.