How much energy is stored in a 2.90-cm-diameter, 14.0-cm-long solenoid that has 200 turns of wire and carries a current of 0.790 ?
Compute the inductance L, in Henries.
The formula you need is here:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html
The stored energy is
E = (1/2) L I^2
E will be in Joules if I is in amperes.
To calculate the energy stored in a solenoid, we can use the formula:
E = (1/2) * L * I^2
where E is the energy stored, L is the inductance of the solenoid, and I is the current.
To find the inductance of the solenoid, we can use the formula:
L = (μ₀ * N² * A) / l
where L is the inductance, N is the number of turns of wire, A is the cross-sectional area of the solenoid, and l is the length of the solenoid.
First, let's calculate the cross-sectional area (A) of the solenoid:
A = π * r²
A = π * (d/2)²
A = π * (2.90 cm / 2)^2
Next, let's convert the diameter and length to meters:
d = 2.90 cm = 0.029 m
l = 14.0 cm = 0.14 m
Now, we can substitute the values into the formula to calculate the cross-sectional area (A):
A = π * (0.029 m / 2)^2
Next, let's calculate the inductance (L) using the formula:
L = (μ₀ * N² * A) / l
where μ₀ is the permeability of free space.
The value of μ₀ is approximately 4π × 10^-7 T·m/A.
Substituting the values into the formula:
L = (4π × 10^-7 T·m/A) * (200 turns)^2 * A / 0.14 m
Now, let's calculate the energy (E) stored in the solenoid using the formula:
E = (1/2) * L * I^2
Substituting the values:
E = (1/2) * L * (0.790 A)^2
Now, let's perform the calculations to find the energy stored in the solenoid.
To calculate the energy stored in a solenoid, we can use the formula for the magnetic energy:
E = 1/2 * L * I^2
Where:
E is the energy stored in the solenoid,
L is the inductance of the solenoid, and
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 (μ₀ ≈ 4π * 10^-7),
N is the number of turns of wire,
A is the cross-sectional area of the solenoid, and
l is the length of the solenoid.
Let's calculate the inductance first:
Given:
Diameter of the solenoid = 2.90 cm = 0.029 m (convert to meters)
Radius of the solenoid, r = 0.029 m / 2 = 0.0145 m
Cross-sectional area of the solenoid, A = π * r^2
Now, let's calculate the cross-sectional area:
A = π * (0.0145)^2
Next, calculate the inductance of the solenoid using the inductance formula:
L = (4π * 10^-7 * (200)^2 * A) / 0.14
Now, substitute the given current value (I = 0.790 A) and the calculated inductance into the energy formula:
E = 1/2 * L * (0.790)^2
Calculate the final answer to find the energy stored in the solenoid.