Two metallic wires X and Y are connected to a series. wire X has length l and radius r . While Y has length 2 l and radius 2 r. find the ratio of total resistance of series combination and resistance wire X , if both the wires are of same material.
To find the ratio of the total resistance of the series combination to the resistance of wire X, we need to calculate the resistances of both wires first.
1. Resistance of wire X:
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.
Given:
Length of wire X = l
Radius of wire X = r
The cross-sectional area of wire X can be calculated using the formula: A = π * r^2
Therefore, the resistance of wire X is: RX = ρ * (l / (π * r^2))
2. Resistance of wire Y:
Given:
Length of wire Y = 2l
Radius of wire Y = 2r
Similarly, calculating the cross-sectional area of wire Y using the formula: A = π * (2r)^2 = 4π * r^2
Therefore, the resistance of wire Y is: RY = ρ * ((2l) / (4π * r^2)) = ρ * (l / (2π * r^2))
3. Total resistance of the series combination:
When resistors are connected in series, the total resistance is the sum of the individual resistances.
Therefore, the total resistance of the series combination is: R_total = RX + RY
Now, we can find the ratio of the total resistance to the resistance of wire X:
Ratio = R_total / RX
= (RX + RY) / RX
= (ρ * (l / (π * r^2)) + ρ * (l / (2π * r^2))) / (ρ * (l / (π * r^2)))
= (l / (π * r^2)) * (1 + 1/2)
= (l / (π * r^2)) * (3/2)
= (3l) / (2π * r^2)
So, the ratio of the total resistance of the series combination to the resistance of wire X is (3l) / (2π * r^2).