Animated Solution for Physics - Magnetic Effects of Current: Comprehension Passage

Imagine a material that can conduct electricity with absolutely zero resistance. No energy loss, no heat generated—just pure, unadulterated electron flow. This isn't science fiction; it's the reality of superconductors. However, this magical state is incredibly fragile. It only exists below a certain temperature, known as the critical temperature, denoted as $T_C$.

But temperature isn't the only enemy of superconductivity. Magnetic fields also play a crucial role. When a superconductor is placed in an external magnetic field $B$, the delicate quantum dance of electrons (known as Cooper pairs) is disrupted. As a result, the material requires an even colder environment to maintain its superconducting state. In other words, the critical temperature $T_C(B)$ is a strictly decreasing function of the magnetic field $B$.

The $R$ vs $T$ Graph

In our first question, we are presented with two different magnetic fields: $B_1$ and $B_2$, with the strict condition that $B_2>B_1$. We need to determine how the resistance $R$ varies with temperature $T$ for both fields.

Because $T_C(B)$ decreases as $B$ increases, a stronger magnetic field will result in a lower critical temperature. Therefore, we can confidently state that:

Let's call these temperatures $T_{C2}$ and $T_{C1}$ respectively. So, $T_{C2}<T_{C1}$.

Now, let's visualize the heating process from absolute zero. At very low temperatures (below $T_{C2}$), both materials are safely in their superconducting states. Their resistance is exactly zero.

As we slowly increase the temperature, we hit $T_{C2}$ first. At this exact moment, the material subjected to the stronger field $B_2$ can no longer sustain superconductivity. Its resistance abruptly jumps from zero to a normal, non-zero value. On the graph, this is represented by the dashed line dropping to zero at the lower temperature $T_{C2}$.

Meanwhile, the material in the weaker field $B_1$ is still perfectly superconducting. It doesn't feel the heat yet! Its resistance remains zero as we continue to warm it up.

Finally, we reach $T_{C1}$. Now, the material in field $B_1$ also loses its superconductivity, and its resistance jumps up to join the normal state curve. This is represented by the solid line dropping to zero at the higher temperature $T_{C1}$.

Looking at the given options, only Option (a) correctly captures this sequence: the dashed line ($B_2$) drops at a lower temperature, and the solid line ($B_1$) drops at a higher temperature.

Bounding the Unknown

In the second question, we are given specific data points to anchor our understanding. We know that in the absence of a magnetic field, the critical temperature is $100\text{ K}$.

We are also told that when a strong field of $7.5\text{ T}$ is applied, the critical temperature drops significantly to $75\text{ K}$.

Our task is to determine the critical temperature when a field of $5\text{ T}$ is applied.

Here is where the power of logical deduction comes in. We don't have the exact mathematical equation for the curve. It could be a parabola, an exponential decay, or any other decreasing function. However, we do know one absolute truth: the function is strictly decreasing.

Since $5\text{ T}$ lies perfectly between $0\text{ T}$ and $7.5\text{ T}$, the critical temperature at $5\text{ T}$ must logically lie between the critical temperatures at those two boundaries.

Mathematically, since $0<5<7.5$, it must be true that:

Substituting our known values into this inequality, we get:

We cannot definitively say that $T_C(5\text{ T})$ is exactly $80\text{ K}$ or any other specific number, because the curve is not necessarily a straight line. The only statement we can make with absolute, undeniable certainty is the inequality itself.

Therefore, Option (b) is the correct and only definite statement.