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 Concept:
A fuse wire is a safety device designed to melt when the current through it exceeds a certain threshold.
The heating process is governed by the Joule effect: \(H = I^2 R t\).
When current \(I\) passes through the wire, electrical energy is converted into thermal energy.
The wire's temperature increases until it reaches its melting point.
The time \(t\) required for the wire to melt depends on the heat produced and the heat required to raise the temperature of the mass of the wire.
Step 2: Key Formula or Approach:
Heat generated: \(H_{gen} = I^2 R t\).
Heat required: \(H_{req} = m \cdot c \cdot \Delta \theta\), where \(m\) is mass, \(c\) is specific heat capacity, and \(\Delta \theta\) is the rise in temperature.
Resistance: \(R = \rho \frac{l}{A} = \rho \frac{l}{\pi r^2}\).
Mass: \(m = \text{density} \times \text{volume} = d \times (\pi r^2 l)\).
Step 3: Detailed Explanation:
Equating the heat generated to the heat required to reach the melting point:
\[ I^2 R t = m c \Delta \theta \]
Substitute the expressions for resistance \(R\) and mass \(m\):
\[ I^2 \left( \rho \frac{l}{\pi r^2} \right) t = (d \cdot \pi r^2 l) c \Delta \theta \]
We are interested in how the time \(t\) depends on the physical dimensions \(r\) and \(l\).
Observe that the length \(l\) appears on both sides of the equation.
Dividing both sides by \(l\):
\[ I^2 \left( \frac{\rho}{\pi r^2} \right) t = (d \cdot \pi r^2) c \Delta \theta \]
This shows that the heating time is independent of the length of the wire, so \(t \propto l^0\).
Now, isolate \(t\) to find its dependence on the radius \(r\):
\[ t = \frac{(d \cdot \pi r^2) c \Delta \theta}{I^2 (\rho / \pi r^2)} \]
\[ t = \frac{d \cdot \pi^2 \cdot c \cdot \Delta \theta}{\rho I^2} r^4 \]
From this final equation, it is clear that for a fixed current \(I\) and fixed material properties (\(d, c, \Delta \theta, \rho\)):
\[ t \propto r^4 \]
Combining the dependencies: \(t \propto r^4 l^0\).
This matches option (C).
Step 4: Final Answer:
The time taken for a fuse wire to heat up to its melting point scales with the fourth power of the radius and is independent of the wire's length.