The magnetic field inside an air-filled solenoid 39 cm long and 2.0 cm in diameter is 0.85 T. Approximately how much energy is stored in this field?

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To calculate the energy stored in the magnetic field of a solenoid, we can use the formula:

E = (1/2) * L * I^2

where:
E is the energy stored in the magnetic field
L is the inductance of the solenoid
I is the current flowing through the solenoid

To find the inductance of the solenoid, we can use the formula:

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 in the solenoid
A is the cross-sectional area of the solenoid
l is the length of the solenoid

We are given:
Length of solenoid (l) = 0.39 m
Diameter of solenoid (d) = 0.02 m
Magnetic field (B) = 0.85 T

First, we need to calculate the number of turns (N) and the cross-sectional area (A) of the solenoid.

To find the number of turns, we can use the formula:

N = (d * π) / p

where:
d is the diameter of the solenoid
π is a mathematical constant (approximately 3.14159)
p is the pitch of the solenoid (distance between adjacent turns)

In this case, the pitch is not given, so we assume a tightly-wound solenoid where each turn is touching the adjacent turn.

N = (0.02 m * π) / 0.02 m = π

Next, we can calculate the cross-sectional area using the formula:

A = (π * d^2) / 4

A = (π * (0.02 m)^2) / 4 = π * 0.0001 m^2

Now, we have all the necessary values to calculate the inductance (L) of the solenoid using the formula mentioned earlier:

L = (4π × 10^−7 T m/A * (π)^2 * 0.0001 m^2) / 0.39 m = (4π^3 × 10^−11 T m²) / 0.39 m ≈ 1.02 × 10^−9 T m²

Finally, we can calculate the energy (E) stored in the magnetic field using the first formula mentioned:

E = (1/2) * L * B^2

E = (1/2) * 1.02 × 10^−9 T m² * (0.85 T)^2 = 0.365 × 10^−9 J ≈ 3.65 nJ

Therefore, the approximate energy stored in the magnetic field of the solenoid is 3.65 nJ.