Understanding Electrical Resistance in a Wire

Electrical resistance is a measure of how strongly a material opposes the flow of electric current. For a conductor like a copper wire, resistance depends on its physical dimensions and the material it is made of.

The formula that describes the resistance (R) of a uniform conductor is given by:

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

Where:

• \( \rho \) (rho) is the resistivity of the material (a property specific to the material, like copper).
• \( l \) is the length of the conductor.
• \( A \) is the cross-sectional area of the conductor.

For a wire with a circular cross-section, the area A is calculated using the radius \( r \):

\( A = \pi r^2 \)

So, the resistance formula can also be written as:

\( R = \rho \frac{l}{\pi r^2} \)

Analyzing the Initial Copper Wire

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

• Radius = \( r \)
• Length = \( l \)

The cross-sectional area of this wire is \( A_1 = \pi r^2 \).

The resistance of this initial wire, let's call it \( R_1 \), is given by the formula:

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

We are told that the resistance of this initial wire is \( R \). So, \( R = \rho \frac{l}{\pi r^2} \).

Analyzing the New Copper Wire

Now, let's consider the second copper wire. It is made of the same material (copper), so its resistivity \( \rho \) is the same as the first wire. We are given its dimensions:

• Radius = \( r_2 = 2r \)
• Length = \( l_2 = l/2 \)

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

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

The resistance of this new wire, let's call it \( R_2 \), is calculated using the resistance formula with its new length and area:

\( R_2 = \rho \frac{l_2}{A_2} = \rho \frac{l/2}{4\pi r^2} \)

Comparing Wire Resistances

Now, we simplify the expression for \( R_2 \):

\( R_2 = \rho \frac{l/2}{4\pi r^2} = \rho \frac{l}{2 \times 4\pi r^2} = \rho \frac{l}{8\pi r^2} \)

We know that the initial resistance \( R = \rho \frac{l}{\pi r^2} \).

Let's rewrite \( R_2 \) to see the relationship with \( R \):

\( R_2 = \frac{1}{8} \times \left( \rho \frac{l}{\pi r^2} \right) \)

Since \( R = \rho \frac{l}{\pi r^2} \), we can substitute R into the equation for \( R_2 \):

\( R_2 = \frac{1}{8} R \)

So, the resistance of the new wire is \( R/8 \).

Summary of Factors Affecting Copper Wire Resistance

This problem demonstrates how resistance depends on the dimensions of the wire. Based on the formula \( R = \rho \frac{l}{\pi r^2} \):

• Resistance is directly proportional to length (\( R \propto l \)). If length increases, resistance increases.
• Resistance is inversely proportional to the cross-sectional area (\( R \propto \frac{1}{A} \)). If area increases, resistance decreases.
• Since area is proportional to the square of the radius (\( A \propto r^2 \)), resistance is inversely proportional to the square of the radius (\( R \propto \frac{1}{r^2} \)). If radius increases, resistance decreases significantly.
• Resistance also depends on the material (resistivity \( \rho \)) and temperature (though not explicitly mentioned in this problem).

In this specific case, doubling the radius increased the area by a factor of \( (2)^2 = 4 \), which tends to decrease resistance by a factor of 4. Halving the length tends to decrease resistance by a factor of 2. The combined effect is a decrease by a factor of \( 4 \times 2 = 8 \).

Resistance Calculation Revision Table

Additional Information on Electrical Properties

Resistivity (\( \rho \)): This is an intrinsic property of a material that quantifies how strongly it resists electric current. Good conductors like copper have low resistivity, while insulators like rubber have high resistivity. Resistivity is temperature-dependent.

Conductivity (\( \sigma \)): Conductivity is the reciprocal of resistivity (\( \sigma = 1/\rho \)). It measures how well a material conducts electricity. Materials with high conductivity have low resistivity.

Effect of Temperature on Resistance: For most conductors (like copper), resistance increases with increasing temperature. This is because thermal vibrations of atoms make it harder for electrons to flow freely. For semiconductors, resistance generally decreases with increasing temperature.