An inductor is in the built in the shape of a solenoid of radius 0.20 cm, length 4.0 cm, with 1000 turns. This inductor is connected to a generator supplying an rms voltage of 1.2×〖10〗^(-4) V AC at an angular frequency of 9.0×〖10〗^3 radians/s.
(a) What is the inductance? Assume that the solenoid can be treated as very long.
(b) What is the rms current flowing through the inductor?
(c) What is the rms energy stored in the inductor?
To find the answers to these questions, we will use the following formulas:
(a) The inductance (L) of a solenoid can be calculated using the formula:
L = (μ₀ * n² * A) / ℓ,
where μ₀ is the permeability of free space (vacuum) equal to 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.
(b) The rms current (I) flowing through an inductor connected to an AC generator can be calculated using Ohm's law and the angular frequency:
I = Vᵣₘₛ / Z,
where Vᵣₘₛ is the rms voltage supplied by the generator and Z is the impedance of the inductor given by:
Z = ωL,
where ω is the angular frequency.
(c) The rms energy (U) stored in an inductor can be calculated using the formula:
U = (1/2) * L * I².
Now let's solve these questions step by step.
(a) In order to find the inductance (L), we need to determine the values of n, A, and ℓ. Given that the solenoid has 1000 turns, a radius of 0.20 cm (0.0020 m), and a length of 4.0 cm (0.04 m), we can calculate the cross-sectional area (A) using the formula:
A = π * r²,
where r is the radius of the solenoid. Therefore,
A = 3.14 * (0.0020 m)² = 1.2566 × 10^(-5) m².
Now we can calculate the inductance (L):
L = (4π × 10^(-7) T·m/A) * (1000 turns)² * (1.2566 × 10^(-5) m²) / (0.04 m).
(b) To find the rms current (I), we need to calculate the impedance (Z) of the inductor. We will use the given angular frequency (ω) of 9.0×10^3 radians/s:
Z = (9.0×10^3 radians/s) * L.
Now we can use Ohm's law to find the rms current (I):
I = (1.2×10^(-4) V) / Z.
(c) Finally, to find the rms energy stored in the inductor (U), we can plug in the values of L and I into the following formula:
U = (0.5) * L * I².