Question

Given below are two statements, one is labelled as Assertion (A) and other one labelled as Reason (R).
Assertion (A) : Soil temperature oscillations over a year penetrate deeper in the soil than over a day.
Reason (R) : Damping depth is proportional to angular frequency or period of oscillations.
In light of the above statements, choose the correct answer from the options given below.

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true but (R) is NOT the correct explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Hint:

Remember the formula: \( d \propto \sqrt{T} \) (Damping depth is proportional to the square root of the period).
Since the annual period is much longer than the daily period, the annual wave penetrates much deeper.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Soil temperature is a dynamic physical property that fluctuates in response to solar radiation inputs at the soil surface.
These fluctuations occur in periodic cycles: a daily (diurnal) cycle driven by the Earth's rotation, and an annual cycle driven by the Earth's revolution around the Sun.
As these temperature waves travel downward into the soil, their amplitude decreases exponentially with depth.
The depth at which the temperature amplitude is reduced to \( 1/e \) (approximately \( 37\% \)) of its surface value is defined as the damping depth (\( d \)).

Step 2: Detailed Explanation:
Let us evaluate the physics behind both statements:
Assertion (A) is true:
Annual soil temperature oscillations (summer to winter) penetrate much deeper into the soil profile than daily oscillations (day to night).
While daily temperature fluctuations are usually negligible below a depth of \( 30\text{ to }50\text{ cm} \), annual temperature variations can be detected down to depths of \( 10\text{ to }15\text{ meters} \) in many soils.
Reason (R) is false:
The mathematical definition of damping depth (\( d \)) in a homogeneous soil is given by:
\[ d = \sqrt{\frac{2K}{\omega}} = \sqrt{\frac{K \cdot T}{\pi}} \]
Where:
\( K \) is the thermal diffusivity of the soil.
\( \omega \) is the angular frequency of the temperature oscillation (\( \omega = \frac{2\pi}{T} \)).
\( T \) is the period of the oscillation.
Looking at the equation:
1. The damping depth (\( d \)) is inversely proportional to the square root of the angular frequency (\( \omega \)).
2. The damping depth (\( d \)) is proportional to the square root of the period of oscillation (\( T \)).
Therefore, the statement “Damping depth is proportional to angular frequency” is scientifically incorrect, as it has an inverse relationship.
Let us compare the periods to see the effect:
For daily oscillations: \( T_{\text{daily}} = 1\text{ day} \) (24 hours).
For annual oscillations: \( T_{\text{annual}} = 365\text{ days} \).
The ratio of their damping depths is:
\[ \frac{d_{\text{annual}}}{d_{\text{daily}}} = \sqrt{\frac{365}{1}} \approx 19.1 \]
This means the annual temperature wave penetrates approximately 19 times deeper than the daily wave because its period is 365 times larger.
Since the Assertion is true but the Reason contains a false mathematical statement, option (C) is the correct choice.

Step 3: Final Answer:
Therefore, the correct choice is that (A) is true but (R) is false.