The resistance (R2) of the wire with a diameter of 1.0 mm is calculated to be 21Ω based on the given proportional relationship.
The calculation is based on the relationship between resistance (R), current (I), and the cross-sectional area (A) of a wire.
1. Cross-sectional Area (A):
The cross-sectional area (A) of a wire with radius (r) is A = πr².
Using the diameter (d)
A = (π/4)d²
2. Comparison of Cross-sectional Areas (A1/A2):
The wire has two different diameters: d1= 0.5mm and d2 = 1mm
The ratio of their cross-sectional areas is A1 / A2 = d1² / d2² = 0.5²mm / 1²mm = 0.25
3. Resistance Proportionality (R ∝ 1 / A):
Resistance (R) of a wire is inversely proportional to its cross-sectional area.
R ∝ 1 / A
4. Equating Resistance Ratios (R2 / R1 = A1 / A2):
If R ∝ 1 / A
The ratio of resistances R2 / R1 = A1 / A2 is equal to the inverse ratio of cross-sectional areas A 1 / A2.
5. Calculating (R2):
Rearrang the equation, (R2 = R1 / (A1 / A2).
Substituting the values,
R2 = 84Ω × 0.25 = 21 Ω
Therefore, the resistance (R2) of the wire with a diameter of 1.0 mm is calculated to be 21Ω based on the given proportional relationship.