If the diameter of a resistance wire is halved then its resistance becomes
Answers
Answer:
4x
Explanation: yes its a bit confusing but if diameter is doubled then basically the area is 1/4 times
so area of cross section is inversely proportional to resistance so now resistance is 4 times.
Hope it was helpful
Given: The resistance of the wire, R1
the length of the wire, L.
the diameter of the wire, d1
To Find: the resistance of the wire when its diameter is halved, R2
Solution:
To calculate R2, the formula used:
- R = ρ x (L / A)
here, R is the resistance of the wire
ρ is the resistivity of the wire
A is the area of cross-section of the wire
L is the length of the wire
Applying the above formula:
For R₁:
R₁ = ρ x (L / A)
A = π x r²
here, r is the radius of the wire
and, radius = diameter / 2
or, r = d / 2
∴ A = π x (d/ 2)²
Putting the value of A in the formula for R:
R₁ = ρ x [L / π x (d/ 2)²]
= ρ x Lx 4 / π x d² ⇒ 1
For R₂, when the diameter of the wire is halved:
R₂ = ρ x (L / A)
As diameter gets halved-
Area, A = π x [(d/2)/ 2]²
A = π x (d/ 2x2)²
Putting the value of A in the formula for R₂:
R₂ = ρ x [ L / π x (d/ 2x2)²]
= ρ x L / (π x d² / 4 x4)
= ρ x L / (π x d² / 16)
= ρ x L x 16 / π x d² ⇒ 2
On dividing the equations 1 and 2:
R₁ / R₂ = (ρ x Lx 4 / π x d²) / (ρ x L x 16 / π x d²)
R₁ / R₂ = 4 / 16
R₁ / R₂ = 1 / 4
Taking recirpocal on both the sides:
R₂ / R₁ = 4 / 1
R₂ = ( 4/1 ) x R₁
R₂ =4 x R₁
R₂ =4R₁
This means new resistance R₂ is four times its original value R₁.
Hence, when the diameter of a wire is halved its resistance becomes four times its original value.