Question

Consider a fuse wire of length \( l \) and radius \( r \). The time of heating (\( t \)) for passing the maximum current will depend on

• A. \( t \propto r^2 l \)
• B. \( t \propto r^{3/2} \)
• C. \( t \propto r^4 l^0 \)
• D. \( t \propto r^{2/3} \)

Hint:

In adiabatic heating of a wire, both the heat capacity and the total resistance scale linearly with length \( l \). As a result, the length factor always cancels out, making the heating time completely independent of the length of the fuse wire.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to find the relation between the heating time \( t \) of a fuse wire and its physical dimensions (length \( l \) and radius \( r \)) when it carries the maximum current.


Step 2: Key Formula or Approach:
We use the conservation of thermal energy, assuming adiabatic heating (no heat loss to the surroundings) during the rapid melting of the fuse:
\[ \text{Heat generated by the current} = \text{Heat required to raise the temperature to the melting point} \]
The resistance of the wire of resistivity \( \rho \), length \( l \), and radius \( r \) is:
\[ R = \frac{\rho l}{\pi r^2} \]
The mass of the wire of density \( d \) is:
\[ m = \pi r^2 l d \]


Step 3: Detailed Explanation:
Let \( I \) be the maximum current, \( s \) be the specific heat capacity, and \( \Delta T \) be the rise in temperature to reach the melting point.
The heat generated in time \( t \) is:
\[ H_{\text{generated}} = I^2 R t = I^2 \left( \frac{\rho l}{\pi r^2} \right) t \]
The heat absorbed by the wire to reach its melting point is:
\[ H_{\text{absorbed}} = m s \Delta T = (\pi r^2 l d) s \Delta T \]
Equating the heat generated to the heat absorbed:
\[ I^2 \left( \frac{\rho l}{\pi r^2} \right) t = (\pi r^2 l d) s \Delta T \]
Solving for the heating time \( t \):
\[ t = \left( \frac{\pi^2 d s \Delta T}{\rho I^2} \right) r^4 \]
From this expression, we observe that:
- The heating time \( t \) is independent of the length \( l \) of the wire (i.e., \( t \propto l^0 \)).
- The heating time \( t \) is proportional to the fourth power of the radius (i.e., \( t \propto r^4 \)).
Thus, \( t \propto r^4 l^0 \).


Step 4: Final Answer:
The time of heating is proportional to \( r^4 l^0 \), which corresponds to option (C).