A copper wire has diameter 0.5 mm and resistivity of 1.6 into 10 power minus 8 ohm metre what will be the length of its resistance 10 ohm how much does the resistance changes the diameter is doubled?

Question

Answer

Diameter = 0.5 mm, radius = 0.25 mm
Area = πr² = π(0.25 × 10⁻³)² = 6.25π × 10⁻⁸
Length = L
Resistitvity, ρ = 1.6 × 10⁻⁸
Resistance, R = ρL/A
10 = (1.6 × 10⁻⁸ × L)/(6.25π × 10⁻⁸)
L = 122.7 m

Doubling the diameter doesn't mean that length remains same. In most cases the question says the wire is 'stretched' or 'compressed'. In such cases the volume remains constant. Volume is nothing but A × L. So, if diameter is doubled, Radius gets doubled and correspondingly area gets 4 times. But since volume is constant it's length decreases by 4 times.
New resistance = (ρL/4)/(4A) = R/16

But in any case if diameter is doubled without change in length then the resistance decrease by 4 times.

Answer

Answer

Calculating Length of Copper Wire

To calculate the length of copper wire, we will use the formula:

Resistance = Resistivity x Length / Area

Where,

Resistivity = 1.6 x 10^-8 ohm metre

Diameter = 0.5 mm

Area = π x (Diameter/2)^2

Resistance = 10 ohm

Now, we can rearrange the equation to solve for length:

Length = Resistance x Area / Resistivity

Substituting the given values, we get:

Length = 10 x π x (0.5/2)^2 / 1.6 x 10^-8

Length = 0.123 km or 123 meters

Calculating Change in Resistance with Doubled Diameter

To calculate the change in resistance with doubled diameter, we will use the formula:

Resistance = Resistivity x Length / Area

Where,

Resistivity = 1.6 x 10^-8 ohm metre

Initial Diameter = 0.5 mm

Final Diameter = 1 mm

Initial Area = π x (0.5/2)^2

Final Area = π x (1/2)^2

Length = 123 meters (calculated in the previous step)

Now, we can calculate the initial resistance:

Initial Resistance = Resistivity x Length / Initial Area

Initial Resistance = 1.6 x 10^-8 x 123 / (π x (0.5/2)^2)

Initial Resistance = 10 ohm

Similarly, we can calculate the final resistance:

Final Resistance = Resistivity x Length / Final Area

Final Resistance = 1.6 x 10^-8 x 123 / (π x (1/2)^2)

Final Resistance = 2.5 ohm

Therefore, the change in resistance is:

Change in Resistance = Final Resistance - Initial Resistance

Change in Resistance = 2.5 - 10

Change in Resistance = -7.5 ohm

This means that the resistance decreases by 7.5 ohm when the diameter is doubled.

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