What happens to the resistance of a wire if its diameter is halved while keeping the length constant?

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Q: What happens to the resistance of a wire if its diameter is halved while keeping the length constant?

Solution: Resistance R is inversely proportional to the area; halving the diameter quarters the area, thus quadrupling the resistance.

Steps: 9

Step 1: Understand that resistance (R) of a wire depends on its dimensions.
Step 2: Know that resistance is inversely proportional to the cross-sectional area (A) of the wire.
Step 3: Remember the formula for the area of a circle: A = π * (d/2)², where d is the diameter.
Step 4: If the diameter is halved, the new diameter is d/2.
Step 5: Calculate the new area with the halved diameter: A_new = π * ((d/2)/2)² = π * (d/4)² = π * (d²/16).
Step 6: Compare the new area to the original area: A_original = π * (d/2)² = π * (d²/4).
Step 7: Notice that A_new is 1/4 of A_original (since d²/16 is 1/4 of d²/4).
Step 8: Since resistance is inversely proportional to area, if the area is reduced to 1/4, the resistance increases by a factor of 4.
Step 9: Conclude that halving the diameter of the wire while keeping the length constant quadruples the resistance.