Question

a fuse wire of circular cross section has a radius of 0.8mm. the wire blows off at a current of 9a. what will be the radius of the wire that will blow off at a current of 1a?

Answer 1

We know that, I2 proportional to R3,
(I2 / I1 )^2 = ( R2 / R1 )^3, R2 = R1 × ( I2 / I1 )^(2 / 3) = 0.8 × (1 / 8)^(2 /3 ) = 0.2 mm

Answer 2

The radius of the wire that will blow off at a current of 1a is 0.18mm.

We know, for a fuse wire of circular cross section, from the heating effect of current, heat produced is

I²Rt = 2Пrlht                                         ...(1)

I is the current flowing,

R is the resistance inside wire,

t is time taken,

r is the radius of cross section,

h is the heat lost by radiation per unit area per unit time.

We know, Resistance (R) = Resistivity(р) × Length(l) ÷ Area (Пr²)

So, R = рl/Пr²

Replacing this value of R in eqn. (1),

I²(рl/Пr²)t = 2Пrlht        

⇒ I²(рl/П)t = 2Пlht×r³

⇒ I² ∝ r³

For I₁ and I₂, we have R₁ and R₂.

So, (I₂/I₁)² = (r₂/r₁)³

From question, we have r₁ = 0.8mm, I₁ = 9A , I₂ = 1A, r₂ = ?

So, (1/9)² = (r₂/0.8)³

⇒ 1/81 = r₂³/0.512

⇒ r₂³ = 0.512/81 = 0.0063

⇒ r₂ = ∛0.0063 = 0.18mm

Required radius is 0.18mm.