Question

A cylindrical wire is stretched to increase its length by 10% calculate the percentage increase in resistance?

Answer 1

The percentage increase in resistance is 21%.

Here is the step-by-step calculation and explanation:

The Key Concepts

1. Resistance Formula: The resistance (R) of a wire is given by:
   $R = \rho \frac{L}{A}$
   where:
   $\rho$ (rho) is the resistivity of the material (it's a constant for the material).
   $L$ is the length of the wire.
   * $A$ is the cross-sectional area of the wire.

2. Constant Volume: When you stretch a wire, you are not adding material. You are just reshaping it. It gets longer, but it must also get thinner. Therefore, its volume ($V$) remains constant.
   $V = L \times A = \text{Constant}$

Step-by-Step Calculation

Let's denote the initial values with a subscript 1 and the final values with a subscript 2.

1. Define the Initial and Final States:

• Initial Length: $L_1$
• Initial Area: $A_1$
• Initial Resistance: $R_1 = \rho \frac{L_1}{A_1}$

2. Define the Change in Length:

The length is increased by 10%. So, the new length ($L_2$) is:
$L_2 = L_1 + (10\% \text{ of } L_1)$
$L_2 = L_1 + 0.10 L_1$
$L_2 = 1.1 L_1$

3. Find the New Area using the Constant Volume Principle:

Since the volume is constant, $V_1 = V_2$.
$L_1 \times A_1 = L_2 \times A_2$

Now, substitute the expression for $L_2$:
$L_1 \times A_1 = (1.1 L_1) \times A_2$

We can cancel $L_1$ from both sides and solve for the new area ($A_2$):
$A_2 = \frac{A_1}{1.1}$
This shows that as the length increases by a factor of 1.1, the area decreases by a factor of 1.1.

4. Calculate the New Resistance:

The new resistance ($R_2$) is:
$R_2 = \rho \frac{L_2}{A_2}$

Now, substitute the new length ($L_2$) and new area ($A_2$):
$R_2 = \rho \frac{1.1 L_1}{A_1 / 1.1}$

Rearranging the terms gives:
$R_2 = \rho \times (1.1 \times 1.1) \frac{L_1}{A_1}$
$R_2 = (1.1)^2 \left( \rho \frac{L_1}{A_1} \right)$

Since $R_1 = \rho \frac{L_1}{A_1}$, we can substitute it back:
$R_2 = (1.1)^2 R_1$
$R_2 = 1.21 R_1$

5. Calculate the Percentage Increase:

The formula for percentage increase is:
$\text{Percentage Increase} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$

$\text{Percentage Increase} = \frac{R_2 - R_1}{R_1} \times 100\%$
Substitute $R_2 = 1.21 R_1$:
$\text{Percentage Increase} = \frac{1.21 R_1 - R_1}{R_1} \times 100\%$
$\text{Percentage Increase} = \frac{0.21 R_1}{R_1} \times 100\%$
$\text{Percentage Increase} = 0.21 \times 100\% = \bf{21\%}$