Question

A copper wire is stretched to make it 0.5% longer. The percentage change in its electrical resistance if its volume remains unchanged is:

Answer 1

formula of resistance is given by, $R=\rho\frac{l}{a}$

where $\rho$ is resistivity or specific resistance.

l is length of wire and a is cross sectional area of wire.

now , question said “volume remains unchanged.”

so, formula of resistance can be converted into $R=\rho\frac{l.l}{a.l}=\rho\frac{l^2}{v}$

where v is volume of wire,

so, it is clear that, resistance of wire is directly proportional to square of length when volume of its remains unchanged.

now, $R\propto l^2$

if % change is very small,

then applying , % change in R = 2 × % change in l

given, the copper wire is stretched to make it 0.5 % longer .

means, % change in length ,l = 0.5 %

so, % change in resistance, R = 2 × 0.5 = 1 %

hence, The percentage change in its electrical resistance is 1 %

Answer 2

The resistance of a wire is given by following equation

R = ρ(L/A)

Where, R = Resistance of wire

ρ = Resistivity of wire

L = length of wire

A = cross sectional area of wire

Volume of wire = cross sectional area x length = A x L

Since 'Resistivity' is a charactersitic property of wire, it does not change with change in length or area. Hence it will remain same for any length of wire.

When the length is stretched to make it 0.5 % longer, its length and cross sectional area will change which will affect the resistance of wire.

New length = L + 0.5 % of L

                    = L+ 0.005 L

                    = 1.005 L

It is mentioned that volume remain unchanged

New volume = New cross sectional area(A') x New length

A x L = New cross sectional area(A') x 1.005 L

Hence, New cross sectional area A' = A/1.005 = 0.995 A

Briefly, features of stretched wire are as follows:

Length (L') = 1.005 L

Cross sectional area (A') = 0.995 A

Resistivity = ρ

Hence Resistance of stretched wire = ρ (L'/A')

                                                            = ρ ( 1.005 L/0.995 A)

                                                             = 1.01 ρ(L/A)

Change in resistance = New resistance - Old resistance

                                     = 1.01 ρ (L/A) - ρ (L/A)

                                    = 0.01 ρ (L/A)

Percentage change = (Change in resistance/Old resistance) x 100

                                  = 0.01 ρ (L/A)/ρ(L/A) x 100

                                  = 0.01 x 100

                                  = 1%