A cylindrical conductor of length L and uniform area of cross section A has resistance R. What will be the area of cross section of another conductor of length 2L and resistance R of the same material?
Answer will be. 2a
To find the area of cross section of another conductor with length 2L and resistance R, we can use the formula for resistance:
R = (ρ * L) / A
where R is the resistance, ρ is the resistivity of the material, L is the length, and A is the area of the cross section.
Since we have the same resistance R and the same material, we can equate the two formulas:
(R1 * L1) / A1 = (R2 * L2) / A2
where R1 is the resistance of the first conductor, L1 is the length of the first conductor, A1 is the area of the cross section of the first conductor, R2 is the resistance of the second conductor, L2 is the length of the second conductor, and A2 is the area of the cross section of the second conductor.
Given that the length of the second conductor is 2L (2 times the length of the first conductor) and the resistance of both conductors is R, we can substitute these values into the equation:
(R * L) / A1 = (R * 2L) / A2
Cancel out the resistance term:
L / A1 = 2L / A2
Cross multiply:
A2 = 2A1
Therefore, the area of cross section of the second conductor, A2, will be twice the area of the cross section of the first conductor, A1.
To find the area of cross section of the other conductor, we can use the formula for resistance:
Resistance (R) = Resistivity (ρ) * Length (L) / Area of cross section (A)
Since both conductors have the same material, the resistivity (ρ) will be the same in both cases. Let's label it as ρ₀.
For the first conductor:
R₁ = ρ₀ * L / A
For the second conductor:
R₂ = ρ₀ * 2L / A₂
Given that R₁ = R₂ = R, we can set up an equation:
R = ρ₀ * L / A = ρ₀ * 2L / A₂
To find the area of cross section for the second conductor (A₂), we can rearrange the equation:
A₂ = (ρ₀ * 2L * A) / (ρ₀ * L)
Simplifying, the ρ₀ and L terms cancel out:
A₂ = 2A
Therefore, the area of cross section for the second conductor is twice the area of the cross section for the first conductor.
since R varies as L/A, changing L to 2L requires that we also change A to 2A, since
L/A = 2L/2A