How does the resistance of a piece of metal change if both its length and its diameter are reduced to one half their original values?
.. im confused because i don't know what the shape of the metal is...please help!
Assume it is a wire (cylinder) and that current flows along the length.
No worries! The resistance of a piece of metal is primarily affected by its dimensions, regardless of its specific shape. In this case, let's assume that the metal is originally in the shape of a cylindrical wire.
To determine how the resistance changes when both the length and diameter are reduced to one half, we need to understand the relationship between resistance, length, and diameter in a cylindrical wire.
The resistance of a wire can be calculated using the formula:
R = ρ * (L / A)
where R is the resistance, ρ is the resistivity of the metal (a characteristic property of the material), L is the length of the wire, and A is the cross-sectional area of the wire.
When both the length (L) and diameter (D) are halved, we can assume that the cross-sectional area (A) remains the same, as the wire is proportional in shape.
Now, let's determine how the length and diameter reduction affects the resistance:
1. Halving the length (L): If the length is halved, the top part of the equation gets divided by 2, resulting in a decrease in resistance.
- R_new = ρ * (L/2) / A
- R_new = (1/2) * (ρ * L) / A
- R_new = (1/2) * R_original
2. Halving the diameter (D): Since the diameter is halved, the cross-sectional area (A) changes. The cross-sectional area is proportional to the square of the diameter. So, when the diameter is halved, A becomes one-fourth of its original value.
- A_new = (π/4) * (D/2)^2
- A_new = (π/4) * (1/4) * D^2
- A_new = (1/16) * A_original
Substituting the new values for length (L) and area (A) into the resistance formula:
R_new = (1/2) * R_original / (1/16)
R_new = 8 * R_original
Therefore, when both the length and diameter of a cylindrical wire are reduced to one half their original values, the resistance of the wire will increase by a factor of 8.
It's essential to note that this explanation assumes the wire is a perfect conductor and follows Ohm's law. Additionally, this applies to cylindrical wires; other shapes may have different formulas to calculate resistance.