Understanding Electrical Resistance of a Wire

The electrical resistance of a cylindrical wire is a measure of how much it opposes the flow of electric current. This resistance depends on several factors, including the material the wire is made of, its length, and its cross-sectional area.

The formula for the resistance (\(R\)) of a wire is given by:

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

Where:

• \(\rho\) (rho) is the resistivity of the material. This is a property of the material itself and indicates how strongly it resists current flow. Since both wires are made of the same material, \(\rho\) will be the same for both.
• \(L\) is the length of the wire.
• \(A\) is the cross-sectional area of the wire. For a cylindrical wire, the cross-sectional area is a circle, so \(A = \pi r^2\), where \(r\) is the radius of the wire.

Calculating Resistance for the First Wire

Let's consider the first cylindrical wire. We are given:

• Length \(L_1 = L\)
• Radius \(r_1 = r\)
• Resistance \(R_1 = R\)

Using the resistance formula, the resistance of the first wire is:

\(R_1 = \rho \frac{L_1}{A_1} = \rho \frac{L}{\pi r^2}\)

So, we have the relationship \(R = \rho \frac{L}{\pi r^2}\).

Calculating Resistance for the Second Wire

Now, let's consider the second wire. We are told that it is made of the same material (so resistivity \(\rho\) is the same) but has:

• Length \(L_2 = \text{twice its length} = 2L_1 = 2L\)
• Radius \(r_2 = \text{twice its radius} = 2r_1 = 2r\)

Let the resistance of the second wire be \(R_2\). Using the resistance formula for the second wire:

\(R_2 = \rho \frac{L_2}{A_2}\)

The cross-sectional area of the second wire is \(A_2 = \pi r_2^2\). Substituting \(r_2 = 2r\):

\(A_2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2\)

Now substitute the values of \(L_2\) and \(A_2\) into the resistance formula for \(R_2\):

\(R_2 = \rho \frac{2L}{4\pi r^2}\)

Comparing the Resistances

We can simplify the expression for \(R_2\):

\(R_2 = \rho \frac{2L}{4\pi r^2} = \rho \frac{1}{2} \frac{L}{\pi r^2}\)

Notice that the term \(\rho \frac{L}{\pi r^2}\) is the resistance of the first wire, which is \(R\). So, we can substitute \(R\) into the equation for \(R_2\):

\(R_2 = \frac{1}{2} \left(\rho \frac{L}{\pi r^2}\right) = \frac{1}{2} R\)

Therefore, the resistance of the second wire is half the resistance of the first wire.

Summary of Calculation

Conclusion on Wire Resistance

When the length of a cylindrical wire is doubled, its resistance tends to double (if the area remains constant). When the radius is doubled, the cross-sectional area becomes four times larger (\(A \propto r^2\)), which tends to reduce the resistance to one-fourth (if the length remains constant). In this specific case, both changes happen simultaneously.

• Length is doubled: resistance increases by a factor of 2.
• Radius is doubled, so area is quadrupled: resistance decreases by a factor of 4.

The combined effect on resistance is multiplication of these factors:

\(R_{\text{new}} = R_{\text{old}} \times (\text{Length Factor}) \times (\text{Area Factor inverse})\)

\(R_2 = R_1 \times (2) \times \left(\frac{1}{4}\right) = R_1 \times \frac{2}{4} = R_1 \times \frac{1}{2}\)

Since \(R_1 = R\), the resistance of the second wire is \(R_2 = \frac{R}{2}\).

Revision Table: Factors Affecting Resistance

Additional Information: Wire Properties and Applications

Understanding wire resistance is crucial in many electrical applications. For example:

• Power Transmission: Wires used for transmitting electricity over long distances are often made thick (large radius/area) to minimize resistance and reduce power loss (\(P = I^2 R\)).
• Heating Elements: Wires with high resistance (often made from specific alloys) are used in heaters and toasters because the power dissipated as heat is proportional to resistance.
• Circuit Design: The resistance of connecting wires in electronic circuits is usually considered negligible compared to components like resistors, but it can become significant in high-current or high-frequency applications, or when using very thin wires.
• Resistivity: Materials like copper and aluminum have low resistivity, making them good conductors for wiring. Materials like nichrome have high resistivity and are used in heating elements.

The relationship \(R = \rho L/A\) is a fundamental concept in electrical engineering and physics, helping us predict and design electrical systems effectively based on the physical dimensions and material properties of conductive components.