Determining the Ratio of Radii for Copper and Iron Wires with Equal Current

When two wires, one made of copper and the other made of iron, are subjected to the same length and potential difference, we often wonder how their individual characteristics, such as the ratio of their radii, affect the current flowing through them. Using Ohm's Law and the formula for electrical resistance, we can derive the exact ratio of the radii that would ensure the same current flows through both wires.

Understanding the Problem

First, let's consider the basic principles of electrical resistance. According to Ohm's Law, the potential difference (V) across a conductor is directly proportional to the current (I) flowing through it, with the constant of proportionality being the resistance (R):

Ohm's Law:

V I R

In the context of wires, we need to consider the resistance formula:

Resistance Formula:

R frac{rho L}{A}

where:

(rho) is the resistivity of the material, 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 given by:

A pi r^2

where r is the radius of the wire.

Setting Up the Equations

Let's denote:

RCu as the resistance of the copper wire RFe as the resistance of the iron wire (rho_{Cu}) and (rho_{Fe}) as the resistivities of copper and iron, respectively.

Using the resistance formula, we can write:

RCu frac{rho_{Cu} L}{pi r_{Cu}^2}

RFe frac{rho_{Fe} L}{pi r_{Fe}^2}

Since the potential difference is the same for both wires and the current must be the same, we can set the resistances equal:

RCu RFe

Substituting the expressions for resistance, we get:

frac{rho_{Cu} L}{pi r_{Cu}^2} frac{rho_{Fe} L}{pi r_{Fe}^2}

Cancelling out the common terms L and (pi) from both sides, we have:

frac{rho_{Cu}}{r_{Cu}^2} frac{rho_{Fe}}{r_{Fe}^2}

Rearranging for the Ratio of Radii

Rearranging the equation to solve for the ratio of the radii, we get:

frac{r_{Cu}^2}{r_{Fe}^2} frac{rho_{Cu}}{rho_{Fe}}

Taking the square root of both sides, we find:

frac{r_{Cu}}{r_{Fe}} sqrt{frac{rho_{Cu}}{rho_{Fe}}}

Resistivity Values

At room temperature:

(rho_{Cu} approx 1.68 times 10^{-8} Omega cdot m) (rho_{Fe} approx 9.71 times 10^{-8} Omega cdot m)

Calculating the Ratio

Substituting these values into the equation, we get:

frac{r_{Cu}}{r_{Fe}} sqrt{frac{1.68 times 10^{-8}}{9.71 times 10^{-8}}} sqrt{frac{1.68}{9.71}} approx sqrt{0.173} approx 0.416

Final Result

Therefore, the ratio of the radii of the copper wire to the iron wire for the same current is approximately:

(frac{r_{Cu}}{r_{Fe}} approx 0.416)

This means that the copper wire must have a radius about 41.6% of the radius of the iron wire to maintain the same current under the same potential difference.

Understanding these concepts is crucial for electrical engineers and students studying electronics. By utilizing Ohm's Law and the formulas for resistance and area, one can accurately determine the necessary dimensions and properties of wires to ensure desired electrical performances.