Question:

A copper wire has diameter 0.5 mm and resistivity of \(1.6×10^{–8} \ Ω m\). What will be the length of this wire to make its resistance 10 Ω? How much does the resistance change if the diameter is doubled?

Updated On: Nov 8, 2023
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Solution and Explanation

Resistance (R) of a copper wire of length l and cross-section A is given by the expression,
\(R = ρ\frac {l}{A}\)
Where,
ρ is resistivity of copper = 1.6 × 10–8 Ω m
R = 10 Ω,
radius of wire r = \(\frac {0.5}{2}\)mm = 0.25 mm = 0.00025 m
𝐴 = 𝜋𝑟
A = 3.14×(0.00025)
A = 0.000000019625 m2
⟹𝑙 = \(\frac {𝑅𝐴}{ρ}\)

⟹𝑙 = \(\frac {10×0.000000019625}{1.6×10^{−8}}\)
⟹𝑙 = 122.72 𝑚
If the diameter (radius) is doubled, the new radius r = 0.5 mm = 0.0005 m
𝐴 = 𝜋𝑟2 
A = 3.14×(0.0005)2
A = 0.000000785 m2 
So, the new resistance will be
\(𝑅^′=ρ\frac {𝑙}{𝐴}\) 

\(R' = \frac {1.6×10^{−8} \times 122.72}{0.000000785 }\)
\(R'=2.5\  Ω\)
Now,
\( \frac {𝑅^′}{𝑅}=\frac {2.5}{10}\)

⟹ \( \frac {𝑅^′}{𝑅}=\frac {1}{4}\)

⟹ \(𝑅^′=\frac 14𝑅\)

Hence, the new resistance will become \(\frac 14\) times the original resistance. 

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Concepts Used:

Resistance

Resistance is the measure of opposition applied by any object to the flow of electric current. A resistor is an electronic constituent that is used in the circuit with the purpose of offering that specific amount of resistance.

R=V/I

In this case,

v = Voltage across its ends

I = Current flowing through it

All materials resist current flow to some degree. They fall into one of two broad categories:

  • Conductors: Materials that offer very little resistance where electrons can move easily. Examples: silver, copper, gold and aluminum.
  • Insulators: Materials that present high resistance and restrict the flow of electrons. Examples: Rubber, paper, glass, wood and plastic.

Resistance measurements are normally taken to indicate the condition of a component or a circuit.

  • The higher the resistance, the lower the current flow. If abnormally high, one possible cause (among many) could be damaged conductors due to burning or corrosion. All conductors give off some degree of heat, so overheating is an issue often associated with resistance.
  • The lower the resistance, the higher the current flow. Possible causes: insulators damaged by moisture or overheating.