A 5.81 10-5 H solenoid is constructed by wrapping 62 turns of wire around a cylinder with a cross-sectional area of 8.20 10-4 m2 When the solenoid is shortened by squeezing the turns closer together, the inductance increases to 8.94 10-5 H. Determine the change in the length of the solenoid.
Answer in m
L₁= μ₀N²A/x₁
=> x₁=μ₀N²A/L₁
L₂= μ₀N²A/x₂
=> x₂= μ₀N²A/L₂
Δx=x₂-x₁=μ₀N²A{(1/L₁)–(1/L₂)}
To find the change in length of the solenoid, we can use the formula for inductance of a solenoid:
L = (μ₀ * N² * A) / l
Where:
L is the inductance
μ₀ is the permeability of free space (4π × 10^(-7) T·m/A)
N is the number of turns
A is the cross-sectional area
l is the length of the solenoid.
Rearranging the formula to solve for length (l):
l = (μ₀ * N² * A) / L
Using the given values:
Initial inductance, L₁ = 5.81 * 10^(-5) H
Final inductance, L₂ = 8.94 * 10^(-5) H
Number of turns, N = 62
Cross-sectional area, A = 8.20 * 10^(-4) m^2
Permeability of free space, μ₀ = 4π × 10^(-7) T·m/A
Plugging in the values:
Initial length, l₁ = (4π × 10^(-7) * 62² * 8.20 * 10^(-4)) / (5.81 * 10^(-5))
Final length, l₂ = (4π × 10^(-7) * 62² * 8.20 * 10^(-4)) / (8.94 * 10^(-5))
The change in length, Δl = l₂ - l₁
To determine the change in length of the solenoid, we can use the formula for inductance:
L = (μ0 * N^2 * A) / l
Where:
L is the inductance,
μ0 is the permeability of free space (4π * 10^-7 T·m/A),
N is the number of turns,
A is the cross-sectional area of the solenoid, and
l is the length of the solenoid.
We are given the initial inductance L1 = 5.81 * 10^-5 H, the final inductance L2 = 8.94 * 10^-5 H, the initial number of turns N1 = 62, and the cross-sectional area A = 8.20 * 10^-4 m^2.
Using the formula, we can rewrite it to solve for l:
l = (μ0 * N^2 * A) / L
First, let's calculate the initial length of the solenoid using the initial values:
l1 = (4π * 10^-7 T·m/A) * (62^2) * (8.20 * 10^-4 m^2) / (5.81 * 10^-5 H)
Next, let's calculate the final length of the solenoid using the final values:
l2 = (4π * 10^-7 T·m/A) * (62^2) * (8.20 * 10^-4 m^2) / (8.94 * 10^-5 H)
Finally, we can find the change in length Δl by subtracting the initial length from the final length:
Δl = l2 - l1
Calculate l1, l2, and Δl to find the change in length of the solenoid in meters.