Understanding Resistance Change: Conductor Length

This solution explains how the electrical resistance of a conductor changes when its length is increased. We will look at the relationship between resistance and length.

Electrical Resistance Basics

Electrical resistance is a measure of how much an object opposes the flow of electric current. It depends on several factors, including the material of the conductor, its length, and its cross-sectional area.

The formula for resistance ($R$) is given by:

R = \rho \frac{L}{A}

Where:

• $R$ is the resistance
• $\rho$ (rho) is the resistivity of the material (an intrinsic property)
• $L$ is the length of the conductor
• $A$ is the cross-sectional area of the conductor

Analyzing Resistance and Length Relationship

From the formula, we can see that resistance ($R$) is directly proportional to the length ($L$) of the conductor, assuming the resistivity ($\rho$) and cross-sectional area ($A$) remain constant.

This can be written as:

R \propto L

This proportionality means that if you increase the length of the conductor, its resistance will increase proportionally, provided other factors are unchanged.

Calculating Resistance Change for Increased Length

Let's consider the initial state of the conductor:

• Initial length = $L_1$
• Initial resistance = $R_1$

According to the direct proportionality, we can write:

R_1 = k L_1

Where '$k$' is a constant of proportionality that includes resistivity and inverse of area (i.e., $k = \rho / A$).

Now, the length of the conductor is increased 3 times. This means the new length ($L_2$) is:

L_2 = 3 \times L_1

The new resistance ($R_2$) will be:

R_2 = k L_2

Substitute the value of $L_2$:

R_2 = k \times (3 L_1)

Rearranging the terms:

R_2 = 3 \times (k L_1)

Since we know that $R_1 = k L_1$, we can substitute this back into the equation:

R_2 = 3 \times R_1

Conclusion on Resistance Change

The calculation shows that the new resistance ($R_2$) is 3 times the original resistance ($R_1$). Therefore, when the length of a conductor is increased 3 times, its resistance becomes three times its original value, assuming the cross-sectional area remains constant.