A nichrome wire of diameter 5mm and length 5m has resistivity of 10^-4 ohm meter. If wire is compressed to double it's diameter. Find the resistance in its new form. Also find the resistance of the wire originally ?
To find the resistance in the new form, we need to consider the change in diameter of the wire.
1. First, let's calculate the original resistance of the wire using the formula: R = (resistivity) * (length) / (cross-sectional area).
Given:
Diameter of the wire (original form) = 5 mm = 0.005 m
Length of the wire = 5 m
Resistivity = 10^-4 ohm meter
To find the cross-sectional area, we need the radius:
Radius (original form) = Diameter (original form) / 2 = 0.005 m / 2 = 0.0025 m
Cross-sectional area (original form) = π * (radius (original form))^2 = π * (0.0025 m)^2 ≈ 0.00001963 m^2
Now we can calculate the original resistance:
R (original form) = (resistivity) * (length) / (cross-sectional area) = (10^-4 ohm meter) * (5 m) / (0.00001963 m^2)
Calculating this, we get:
R (original form) ≈ 2.55 ohms
2. Next, let's calculate the new resistance after compressing the diameter to twice its original size.
Diameter of the wire (new form) = 2 * 0.005 m = 0.01 m
To find the new cross-sectional area, we need the new radius:
Radius (new form) = Diameter (new form) / 2 = 0.01 m / 2 = 0.005 m
Cross-sectional area (new form) = π * (radius (new form))^2 = π * (0.005 m)^2 ≈ 0.00007854 m^2
Now we can calculate the new resistance:
R (new form) = (resistivity) * (length) / (cross-sectional area) = (10^-4 ohm meter) * (5 m) / (0.00007854 m^2)
Calculating this, we get:
R (new form) ≈ 0.636 ohms
Therefore, the resistance in the new form is approximately 0.636 ohms, and the resistance of the wire originally (in its original form) is approximately 2.55 ohms.
To find the resistance in the new form of the wire, we need to consider two factors: the change in length and the change in cross-sectional area.
Step 1: Calculate the original resistance of the wire.
The resistance of a wire can be calculated using the formula: R = (ρ * L) / A
where R is the resistance, ρ is the resistivity, L is the length of the wire, and A is the cross-sectional area of the wire.
Given:
- Diameter of the wire = 5 mm = 0.005 m
- Length of the wire = 5 m
- Resistivity of the wire = 10^-4 ohm meter
Step 1.1: Calculate the original cross-sectional area of the wire.
The original cross-sectional area (A) can be calculated using the formula: A = π * (r^2)
where r is the radius of the wire, which is half the diameter.
Radius (r) = (1/2) * diameter = (1/2) * 0.005 m = 0.0025 m
Plugging in the values:
A = π * (0.0025^2) = 0.00001963 m^2
Step 1.2: Calculate the original resistance.
R = (ρ * L) / A = (10^-4 * 5) / 0.00001963 = 0.025 ohms
So, the resistance of the wire in its original form is 0.025 ohms.
Step 2: Calculate the resistance in the new form of the wire.
In the new form, the diameter of the wire is doubled while the length remains the same.
Step 2.1: Calculate the new cross-sectional area of the wire.
The new cross-sectional area, A', can be calculated using the same formula as before: A' = π * (r'^2)
The new radius, r', is twice the original radius, r.
r' = 2 * r = 2 * 0.0025 m = 0.005 m
Plugging in the values:
A' = π * (0.005^2) = 0.00007854 m^2
Step 2.2: Calculate the new resistance.
Using the formula: R' = (ρ * L) / A'
R' = (10^-4 * 5) / 0.00007854 = 0.636 ohms
So, the resistance of the wire in its new form is 0.636 ohms.