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...

Key Concepts

Electrical Resistivity

Electrical resistivity is an intrinsic material property that quantifies how strongly a material opposes the flow of electric current. It is a crucial parameter in determining the resistance of a conductor given its length and cross-sectional area. Understanding resistivity is key when comparing conductors made from different materials, such as copper and aluminum, to achieve a desired overall resistance.

Resistance of a Conductor

The electrical resistance of a conductor is determined by the resistivity of the material, the length of the conductor, and its cross-sectional area. The relationship is given by the formula R = (?L)/A, where R is resistance, ? is resistivity, L is the length, and A is the cross-sectional area. This concept allows for designing wires with specific dimensions to match desired resistance values across different materials.

Relationship Between Diameter, Cross-Sectional Area, and Resistance

For wires with circular cross-sections, the diameter directly influences the cross-sectional area, which in turn affects the resistance since the area is proportional to the square of the diameter (A = ?d²/4). In cases where wires of different materials must have equal resistance for the same length, it is essential to adjust the diameter appropriately to account for differences in material resistivity while maintaining the required conductive properties.

Transcript

00:01 So we're given that aluminum wire has a diameter of 2 .14 millimeters or 2 .14 times 10 and minus 3 meters.

00:11 And we're also told that a copper wire has the same resistance in length as the aluminum wire.

00:19 So to distinguish between the resistance, i just put cu for copper and al for aluminum.

00:29 So this equation says that they have the same resistance, and this equation says that they have the same length.

00:37 And we're asked to find what diameter of the copper wire will satisfy the same resistance and length of aluminum wire with a diameter of 2 .14 millimeters.

00:51 So i'll actually solve this on the next page and we'll use equation 25 .10, which tells us that the resistance is the resistivity times the length over a cross -actional area.

01:18 And then we'll use our resistance of copper is equal to the resistance.

01:27 Of aluminum because we're told that both we want to find the copper wire with the same resistance as the aluminum wire so if we substitute this portion into this equation then we find that the resistivity for copper times the length of copper divided by the cross -sectional area of copper is equal to resistivity of aluminum, length of aluminum, over cross -actual area of aluminum.

02:21 But we're also told that they have the same length.

02:30 So that means that we can get rid of these two lengths because they're the same.

02:35 So then we find that the resistivity of copper over the cross -sectional area of copper is equal to resistivity of aluminum over cross -sectional area of aluminum.

02:55 And i got this by simply cancelling out these two because they're the same thing.

03:04 And now remember that the cross -sectional area is is corresponds to a circle...

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