a wire of lengh 2meter has a resistance of 4 ohms,obrain the resistance if specific resistance is doubled,diameter is doubled and the length is made three times of the first.
L1=2m
R1=4ohms
D=2d
L2=3×2=6m
A1=πd2\4=(πr2)
A2=π(2d)2/4=π4d2/4
R₁=ρ₁L₁/A₁=ρ₁L₁/(πd₁²/4)
R₂=ρ₂L₂/A₂=ρ₂L₂ /(πd₂²/4)
R₂/R₁=ρ₂L₂ πd₁²/ρ₁L₁πd₂² =
=2ρ₁3L₁(d₁)²/ρ₁L₁(2d₁)² =2•3•/4= 3/2=1.5
R₂ = 1.5R₁ = 1.5•4 = 6 Ω
To find the new resistance, we need to consider the changes in the specific resistance, diameter, and length of the wire.
1. Specific Resistance: If the specific resistance is doubled, it means the wire's resistance per unit length has increased.
2. Diameter: If the diameter is doubled, we can use the formula for resistance in a wire, which states that resistance is inversely proportional to the cross-sectional area of the wire. Since the diameter is doubled, the cross-sectional area quadruples (since area is proportional to the square of the diameter).
3. Length: If the length is made three times the original length, the wire will have a greater amount of material, thus increasing its resistance.
Let's solve this step by step:
Step 1: Calculate the new specific resistance.
Since the specific resistance is doubled, let's call the new specific resistance "r". Thus, r = 2 * 4 = 8 ohms.
Step 2: Calculate the new cross-sectional area.
Since the diameter is doubled, the area increases by a factor of 4 (2^2). Let's call the new cross-sectional area "A". If the original area is "A_0," then A = 4 * A_0.
Step 3: Calculate the new length.
Since the length is made three times the original length, let's call the new length "L." Thus, L = 3 * 2 = 6 meters.
Step 4: Calculate the new resistance.
The resistance of a wire is given by the formula: R = (ρ * L) / A, where ρ represents the specific resistance, L represents the length, and A represents the cross-sectional area.
Substituting the values into the formula: R = (8 ohms * 6 meters) / (4 * A_0).
However, since we know that A = 4 * A_0, we can simplify the expression: R = (8 * 6) / (4 * A_0).
Rearranging further, we find: R = 12 / A_0.
Therefore, the new resistance is 12 / A_0 ohms.