The resistance of a copper wire of diameter 10mm and length 2m is 15 Ω. Determine the new resistance if
i) the length is doubled and the diameter halved
ii) the length increases by three fold and the diameter doubled
iii) the length decreases by two fold and the diameter is halved
iv) the length increases by four fold and the diameter is trebled
To determine the new resistance in each scenario, we need to use the formula for the resistance of a wire:
Resistance = (ρ x Length) / (Area)
where:
- ρ is the resistivity of the material (which is constant for copper),
- Length is the length of the wire, and
- Area is the cross-sectional area of the wire.
We'll address each scenario step-by-step:
i) When the length is doubled and the diameter is halved:
1. Let's assume the original length is L1 and the original diameter is D1.
2. The new length is twice the original length, so the new length L2 = 2L1.
3. The new diameter is half the original diameter, so the new diameter D2 = D1/2.
Now, let's calculate the new resistance using the formula:
Resistance2 = (ρ x L2) / (A2)
4. Since the area of a circle is proportional to the square of its radius, and the radius is half the diameter, the new area A2 can be calculated as A2 = π x (D2/2)^2.
5. Substituting the given values, we can calculate the new resistance:
Resistance2 = (ρ x 2L1) / ( π x (D1/2)^2 )
ii) When the length increases by three-fold and the diameter is doubled:
1. Assuming the original length is L1 and the original diameter is D1.
2. The new length is three times the original length, so the new length L2 = 3L1.
3. The new diameter is twice the original diameter, so the new diameter D2 = 2D1.
Now, let's calculate the new resistance using the formula:
Resistance2 = (ρ x L2) / (A2)
4. Calculate the new area A2 as A2 = π x (D2/2)^2.
5. Substitute the given values into the formula to calculate the new resistance:
Resistance2 = (ρ x 3L1) / ( π x (2D1/2)^2 )
iii) When the length decreases by two-fold and the diameter is halved:
1. Assume the original length is L1 and the original diameter is D1.
2. The new length is half the original length, so the new length L2 = L1/2.
3. The new diameter is half the original diameter, so the new diameter D2 = D1/2.
Now, let's calculate the new resistance using the resistance formula:
Resistance2 = (ρ x L2) / (A2)
4. Calculate the new area A2 as A2 = π x (D2/2)^2.
5. Substitute the given values into the formula to calculate the new resistance:
Resistance2 = (ρ x L1/2) / ( π x (D1/2/2)^2 )
iv) When the length increases by four-fold and the diameter is trebled:
1. Assume the original length is L1 and the original diameter is D1.
2. The new length is four times the original length, so the new length L2 = 4L1.
3. The new diameter is three times the original diameter, so the new diameter D2 = 3D1.
Now, let's calculate the new resistance using the resistance formula:
Resistance2 = (ρ x L2) / (A2)
4. Calculate the new area A2 as A2 = π x (D2/2)^2.
5. Substitute the given values into the formula to calculate the new resistance:
Resistance2 = (ρ x 4L1) / ( π x (3D1/2)^2 )
These are the step-by-step calculations for each scenario to determine the new resistance.
To calculate the new resistance in each scenario, we need to use the formula for the resistance of a wire:
R = (ρ * L) / A
where:
R is the resistance
ρ is the resistivity of the material (in this case, copper)
L is the length of the wire
A is the cross-sectional area of the wire
First, we need to determine the original cross-sectional area of the wire using the given diameter.
i) Scenario 1 - Length doubled and diameter halved:
1. Since the diameter is halved, the radius will be reduced to half as well (5mm or 0.005m).
2. The cross-sectional area is given by the formula: A = π * r^2. Substituting the values, we get: A = π * (0.005)^2.
3. Now, we can calculate the new resistance using the new length (4m) and the calculated cross-sectional area. Plug in these values in the formula R = (ρ * L) / A.
ii) Scenario 2 - Length increases by three-fold and diameter doubled:
1. The new length is 2m * 3 = 6m.
2. Since the diameter is doubled, the radius will also be multiplied by 2, giving us a new radius of 0.01m.
3. Calculate the new cross-sectional area using the formula A = π * r^2.
4. Substitute the new length and cross-sectional area in the formula R = (ρ * L) / A to find the new resistance.
iii) Scenario 3 - Length decreases by two-fold and diameter halved:
1. The new length is 2m / 2 = 1m.
2. Since the diameter is halved, the radius is reduced to half as well (5mm or 0.005m).
3. Calculate the new cross-sectional area using the formula A = π * r^2.
4. Plug in the new length and cross-sectional area in the formula R = (ρ * L) / A to find the new resistance.
iv) Scenario 4 - Length increases by four-fold and diameter trebled:
1. The new length is 2m * 4 = 8m.
2. Since the diameter is trebled, the radius is multiplied by 3, giving us a new radius of 0.03m.
3. Calculate the new cross-sectional area using the formula A = π * r^2.
4. Substitute the new length and cross-sectional area in the formula R = (ρ * L) / A to find the new resistance.
By following these steps and applying the formulas, you should be able to determine the new resistance in each scenario.