What diameter must a copper wire have if its resistance is to be the same as that of an equal le

What diameter must a copper wire have if its resistance is...

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Key Concepts

Resistivity

Resistivity is an intrinsic property of a material that quantifies how strongly it opposes the flow of electric current. It is used to determine how the dimensions and material composition of a conductor influence its resistance. Lower resistivity means the material conducts electricity better, which is why different materials require adjustments in geometric dimensions like diameter to achieve the same resistance.

Electrical Resistance in Conductors

The resistance of a conductor depends on its resistivity, length, and cross-sectional area, as described by the formula R = ?L/A. This relationship is crucial in comparing conductors made of different materials over the same length, since a change in material (with a different resistivity) will require an adjustment in the cross-sectional area—often expressed in terms of diameter for wires—to maintain the same resistance.

Cross-Sectional Area and Diameter Relationship

For circular wires, the cross-sectional area is directly related to the square of the diameter through the equation A = ?(d^2)/4. Understanding this relationship allows one to determine the necessary diameter of a conductor made of a different material so that, when combined with the material's resistivity, the desired electrical resistance is achieved.

Transcript

00:01 We want to know what diameter must a copper wire be, copper wire having a resistivity of 1 .72 times 10 to the negative 8 omm meters.

00:12 So that is the same as a aluminum wire.

00:17 So the resistivity of aluminum is 2 .75 times 10 to the negative 8 omec meters.

00:27 And it has a diameter of 3 .84 millimeters.

00:33 And they're going to have the same length.

00:37 And so our equation for this would be resistance is equal to resistivity times length divided by area.

00:46 And so this is resistivity of one of them, and they're going to be the same.

00:51 And so we're going to have resistance of copper, resistance of aluminum.

01:00 And so we can set these equal to each other because they will be the same.

01:03 The resistivity of copper times the length.

01:08 Area is going to be pi r squared.

01:11 It's equal to resistivity of the luminal times the length divided by pi r squared.

01:18 And so we can divide both sides by length and multiply both sides by pi.

01:25 And so we get the resistivity of copper divided by radius squared and this is the radius of the copper is equal to.

01:33 To resistivity of aluminum divided by the radius of aluminum squared.

01:41 And so we want to know the diameter of the copper...

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