A certain length of wire has a resistance of 21 ohms. This wire is melted down and from the same volume of material a new wire is formed that is three times longer than the original wire. What is the resistance of a new wire?
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To solve this problem, we need to understand the relationship between the length of a wire and its resistance. Resistance is typically calculated based on the length of the wire, the material's resistivity, and the wire's cross-sectional area.
In this scenario, we know that the original wire has a resistance of 21 ohms. Let's assume the original length of the wire is 'L' units, and its resistance is 'R'. Therefore, we have the equation R = 21 ohms.
Next, we are informed that the new wire is three times longer than the original wire. Let's denote the length of the new wire as '3L' (since it is three times longer). We need to calculate the resistance of this new wire.
To find the resistance, we need to consider the formula for resistance, which involves resistivity, length, and cross-sectional area. Since the material and cross-sectional area remain the same, we can assume they will cancel out in the formula.
The formula for resistance is:
R = (resistivity × length) / cross-sectional area
Since the material and cross-sectional area remain the same, we can say that the resistivity × cross-sectional area is constant for both the original and new wire. So we can write the equation for the original wire as R1 = (k × L), and for the new wire as R2 = (k × 3L), where 'k' represents the constant term.
Now, we can set up the proportion:
R1 / L = R2 / 3L
Substituting the values of R1 and R2, we have:
(21 ohms) / L = (k × 3L) / (3L)
Simplifying the equation:
21 ohms = k
Hence, the resistivity × cross-sectional area (k × cross-sectional area) is equal to 21 ohms.
Now, let's find the resistance of the new wire, R2:
R2 = (k × 3L) / 3L
Canceling out the '3L' terms:
R2 = k
Therefore, the resistance of the new wire is equal to the resistivity × cross-sectional area, which we found to be 21 ohms.
In conclusion, the resistance of the new wire is 21 ohms.
To find the resistance of the new wire, we need to use the formula for resistance, which is:
Resistance = resistivity x (length / cross-sectional area)
Given that the volume of the wire remains the same, we know that:
Volume = length x cross-sectional area
Since the new wire is three times longer than the original wire, the length of the new wire is 3 times the length of the original wire.
Let's denote the original length as L and the new length as 3L.
Since the volume remains the same, we can set up the following equation:
L x cross-sectional area = 3L x new cross-sectional area
The lengths cancel out, giving us:
cross-sectional area = 3 x new cross-sectional area
Now, let's denote the original resistance as R and the resistance of the new wire as Rnew.
We are given that the original resistance is 21 ohms, so R = 21 ohms.
Plugging these values into the resistance formula:
Rnew = resistivity x (3L / (3 x new cross-sectional area))
Since the resistivity and length of the wire remain the same, we can simplify the equation to:
Rnew = (3 / 3) x R
Rnew = R
Therefore, the resistance of the new wire is also 21 ohms.
Area L = volume
so if 3 times longer, then 1/3 the area
3 * 21 = 63 ohms