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A uniform resistance wire of length L and diameter d has a resistance R. Another wire of same material has length 4 L and diameter 2 d, the resistance will be
- a)2 R
- b)R
- c)R/2
- d)R/4
Correct answer is option 'B'. Can you explain this answer?
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To understand why the resistance of the second wire is the same as the resistance of the first wire, we need to consider the factors that determine the resistance of a wire.
The resistance of a wire is given by the formula:
R = ρ * (L/A)
where R is the resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
1. Resistance of the first wire:
Let's assume the resistivity of the wire material is constant.
For the first wire:
Length (L) = L
Diameter (d) = d
The cross-sectional area of the wire can be calculated using the formula for the area of a circle:
A = π * (d/2)^2 = π * (d^2/4)
Substituting the values into the resistance formula, we get:
R1 = ρ * (L/A) = ρ * (L/(π * (d^2/4)))
2. Resistance of the second wire:
For the second wire:
Length (L) = 4L (four times the length of the first wire)
Diameter (d) = 2d (twice the diameter of the first wire)
The cross-sectional area of the wire can be calculated using the formula for the area of a circle:
A = π * ((2d)/2)^2 = π * (d^2)
Substituting the values into the resistance formula, we get:
R2 = ρ * (4L/π * (d^2))
3. Comparing the resistances:
To compare the resistances of the two wires, we need to simplify the expression for R2 and see if it is equal to R1.
R2 = ρ * (4L/π * (d^2))
= ρ * (4L/(π * (d^2/4)) * (4/4))
= ρ * (4L/(π * (d^2/4)) * (16/16))
= ρ * (16L/(π * d^2))
From the above expression, we can see that R2 is equal to R1.
Therefore, the resistance of the second wire is the same as the resistance of the first wire, which is option B.
The resistance of a wire is given by the formula:
R = ρ * (L/A)
where R is the resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
1. Resistance of the first wire:
Let's assume the resistivity of the wire material is constant.
For the first wire:
Length (L) = L
Diameter (d) = d
The cross-sectional area of the wire can be calculated using the formula for the area of a circle:
A = π * (d/2)^2 = π * (d^2/4)
Substituting the values into the resistance formula, we get:
R1 = ρ * (L/A) = ρ * (L/(π * (d^2/4)))
2. Resistance of the second wire:
For the second wire:
Length (L) = 4L (four times the length of the first wire)
Diameter (d) = 2d (twice the diameter of the first wire)
The cross-sectional area of the wire can be calculated using the formula for the area of a circle:
A = π * ((2d)/2)^2 = π * (d^2)
Substituting the values into the resistance formula, we get:
R2 = ρ * (4L/π * (d^2))
3. Comparing the resistances:
To compare the resistances of the two wires, we need to simplify the expression for R2 and see if it is equal to R1.
R2 = ρ * (4L/π * (d^2))
= ρ * (4L/(π * (d^2/4)) * (4/4))
= ρ * (4L/(π * (d^2/4)) * (16/16))
= ρ * (16L/(π * d^2))
From the above expression, we can see that R2 is equal to R1.
Therefore, the resistance of the second wire is the same as the resistance of the first wire, which is option B.
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