Compare the total magnetic energy in two solenoids, each with N turns, area A, and current I, but length x and 2x. Denote Ux the total magnetic energy in the solenoid with length x, and U2x the one in the solenoid with length 2x. What is U2x/Ux?
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To compare the total magnetic energy in two solenoids with different lengths, we can use the formula for the magnetic energy in a solenoid. The formula for the magnetic energy in a solenoid is given by:
U = (μ₀/2) * N² * A * I²
Where:
U is the total magnetic energy
μ₀ is the permeability of free space (a constant)
N is the number of turns in the solenoid
A is the cross-sectional area of the solenoid
I is the current flowing through the solenoid
Now, we want to compare the magnetic energies for two solenoids with lengths x and 2x, respectively.
For the solenoid with length x, let's denote its magnetic energy as Ux.
For the solenoid with length 2x, let's denote its magnetic energy as U2x.
We can write the equations for Ux and U2x as follows:
Ux = (μ₀/2) * N² * A * I² * (x)
U2x = (μ₀/2) * N² * A * I² * (2x)
Now, let's find the ratio U2x/Ux:
U2x/Ux = [(μ₀/2) * N² * A * I² * (2x)] / [(μ₀/2) * N² * A * I² * (x)]
Notice that some terms cancel out:
U2x/Ux = (2x) / (x)
Simplifying further:
U2x/Ux = 2
Therefore, the ratio U2x/Ux is equal to 2.
This means that the total magnetic energy in the solenoid with double the length is twice as large compared to the solenoid with the shorter length.