The resistance of a wire of cross-section ‘a’ and length ‘l’ is R ohm. The resistance of another wire of the same material and of the same length but cross-section ‘4a’ will be then x
Question
The resistance of a wire of cross-section ‘a’ and length ‘l’ is R ohm. The resistance of another wire of the same material and of the same length but cross-section ‘4a’ will be then x
Solution 1
The resistance (R) of a wire can be calculated using the formula:
R = ρl/a
where: ρ is the resistivity of the material, l is the length of the wire, and a is the cross-sectional area of the wire.
Given that the resistance of the first wire is R ohm, we can write:
R = ρl/a
The resistance of the second wire (let's call it R2) with the same material and length but with a cross-sectional area of '4a' can be calculated using the same formula:
R2 = ρl/(4a)
Now, we can express R2 in terms of R. We know that R = ρl/a, so we can substitute this into the equation for R2:
R2 = R/4
So, the resistance of the second wire is one-fourth the resistance of the first wire. Therefore, x = 1/4.
Solution 2
The resistance (R) of a wire is given by the formula:
R = ρl/a
where: ρ is the resistivity of the material, l is the length of the wire, and a is the cross-sectional area of the wire.
Given that the resistance of the first wire is R ohm, the length is 'l', and the cross-sectional area is 'a', we can write:
R = ρl/a
Now, for the second wire, the length is the same ('l') but the cross-sectional area is '4a'. Substituting these values into the formula, we get:
R' = ρl/(4a)
Since ρ and l are constants, we can see that the resistance of the second wire is one-fourth that of the first wire. Therefore, the resistance of the second wire (R') is R/4 ohm.
So, x = R/4.
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