a wire has a resistance of 1 ohm if the length is increased four times and the area is double the original, find the new resistance of the wire.
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Identity problems?
To find the new resistance of the wire, we need to consider the effect of changing the length and the area on the resistance. Let's break it down step-by-step:
1. Resistance (R) is given by the formula: R = (ρ * L) / 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.
2. In this case, the old resistance is 1 ohm, and we want to find the new resistance.
3. Given that the length of the wire is increased four times, we can say that L2 = 4 * L1, where L1 is the original length of the wire and L2 is the new length.
4. Also, the area is double the original, so we can say that A2 = 2 * A1, where A1 is the original area of the wire and A2 is the new area.
5. Now, let's substitute the new values into the resistance formula:
R2 = (ρ * L2) / A2
= (ρ * (4 * L1)) / (2 * A1)
= (2 * ρ * L1) / A1
6. Since the resistivity (ρ) remains the same, we can say that R2 = 2 * R1, where R1 is the original resistance of the wire and R2 is the new resistance.
Therefore, the new resistance of the wire is twice the original resistance.
To find the new resistance of the wire, we need to consider how changes in length and area affect the resistance.
Resistance is directly proportional to the length of the wire and inversely proportional to the cross-sectional area. Mathematically, the resistance of a wire can be expressed as:
R = ρ * (L/A)
Where R is the resistance, ρ (rho) is the resistivity of the material (a constant), L is the length of the wire, and A is the cross-sectional area of the wire.
Given that the original resistance is 1 ohm, we can substitute this value into the equation and solve for ρ:
1 = ρ * (L/A)
Now, let's consider the changes in length and area. The length is increased four times, so the new length (L') will be 4L. The area is doubled, so the new area (A') will be 2A.
Substituting these values into the equation, we get:
1 = ρ * (4L/2A)
Simplifying further:
1 = 2ρ * (L/A)
Now, we can see that the new resistance (R') is equal to 2 times the original resistance:
R' = 2 * R = 2 * 1 ohm = 2 ohms
Therefore, the new resistance of the wire is 2 ohms.