The resistance(r)of a wire varies directly as the length(L)of the wire and inversely as the square of the diameter(d). If the resistance is 9ohms in 50ft of wire that has a diameter of 0.05in, find the resistance in 50ft of a similar wire that has a diameter of 0.02in.
Resistance=constant*Length/diameter^2
Two ways you can work this:
1) In the equation, solve for the constant (you have resistance 9 ohms, Length 50, and diameter .05). Then use that constant in the equation to solve for the new conditions,
2) Notice the length is the same, however...
Resistance1=constant*Length1/diameter1^2
and
Resistance2=constant*Length2/diameter2^2
now dividing the second equation by the first, and noting that the length1 and length 2 are the same..
Resistance2/resistance1=(diameter2/diameter1)^2
or resistance2=resistance1(diameter2/diameter1)^2
To solve this problem using the first method, we need to first find the constant of proportionality in the equation.
Given:
Resistance1 = 9 ohms
Length1 = 50 ft
Diameter1 = 0.05 in
We can substitute these values into the equation and solve for the constant:
9 = constant * 50 / (0.05)^2
9 = constant * 50 / 0.0025
9 = constant * 20,000
Dividing both sides by 20,000, we get:
constant = 0.00045
Now that we have the constant, we can use it to find the resistance in the second scenario where the diameter is 0.02 in.
Given:
Length2 = 50 ft
Diameter2 = 0.02 in
Using the equation, we can substitute the values:
Resistance2 = 0.00045 * 50 / (0.02)^2
Resistance2 = 0.00045 * 50 / 0.0004
Resistance2 = 0.045 * 50
Resistance2 = 2.25 ohms
Therefore, the resistance in 50 ft of a similar wire with a diameter of 0.02 in is 2.25 ohms.
Alternatively, using the second method, we can use the ratio of diameters to find the new resistance.
Given the same values as before:
Diameter1 = 0.05 in
Diameter2 = 0.02 in
We can calculate the ratio of diameters squared:
(diameter2 / diameter1)^2 = (0.02 / 0.05)^2 = (0.4)^2 = 0.16
Now, we can multiply this ratio by the original resistance:
Resistance2 = Resistance1 * (diameter2 / diameter1)^2
Resistance2 = 9 * 0.16
Resistance2 = 1.44 ohms
Using either method, we find that the resistance in 50 ft of a similar wire with a diameter of 0.02 in is 2.25 ohms or 1.44 ohms, respectively.