A solenoid of radius 2.30 cm has 380 turns and a length of 15.0 cm.
(a) Find its inductance.
(b) Find the rate at which current must change through it to produce an emf of 76.0 µV
To find the inductance of the solenoid, we can use the formula:
L = (μ_0 * N^2 * A) / l
where:
L = inductance of the solenoid
μ_0 = permeability of free space (4π x 10^-7 T·m/A)
N = number of turns
A = cross-sectional area of the solenoid
l = length of the solenoid
(a) Finding the inductance:
Given:
Radius of the solenoid (r) = 2.30 cm = 0.023 m
Number of turns (N) = 380
Length of the solenoid (l) = 15.0 cm = 0.15 m
Step 1: Calculate the cross-sectional area of the solenoid.
A = π * r^2
A = π * (0.023)^2
A = π * 0.000529
A ≈ 0.001663 m^2
Step 2: Plug the given values into the inductance formula.
L = (μ_0 * N^2 * A) / l
L = (4π x 10^-7 * 380^2 * 0.001663) / 0.15
L ≈ 0.018 H
Therefore, the inductance of the solenoid is approximately 0.018 H.
(b) To find the rate at which the current must change through the solenoid to produce an emf, we can use Faraday's Law of electromagnetic induction:
ε = -L * (dI/dt)
where:
ε = magnitude of the induced emf
L = inductance of the solenoid
(dI/dt) = rate of change of current
Given:
Induced emf (ε) = 76.0 µV = 76.0 x 10^-6 V
Step 1: Rearrange the equation to solve for (dI/dt).
(dI/dt) = -ε / L
(dI/dt) = -(76.0 x 10^-6) / 0.018
(dI/dt) ≈ -4.22 A/s
Therefore, the rate at which the current must change through the solenoid to produce an emf of 76.0 µV is approximately -4.22 A/s. The negative sign indicates that the current must decrease at this rate.
To find the inductance of a 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)
N is the number of turns
A is the cross-sectional area of the solenoid
l is the length of the solenoid
Let's calculate the inductance for the given solenoid:
(a) To find the inductance, we need to find the cross-sectional area of the solenoid first.
The formula for the area of a circle is:
A = π * r²
where:
A is the area
r is the radius of the circle
Given that the radius of the solenoid is 2.30 cm, we can calculate the cross-sectional area:
A = π * (2.30 cm)²
Now, convert the radius to meters because the permeability of free space is given in SI units (meters):
A = π * (0.023 m)²
Calculate the value for A to proceed further.
(b) Once we have the inductance, we can move on to finding the rate at which current must change through the solenoid to produce an emf.
The formula for induced emf is given by:
ε = -L * (dI / dt)
where:
ε is the induced emf
L is the inductance
dI/dt is the rate of change of current
Given that the induced emf is 76.0 µV (76.0 × 10^-6 V), we can rearrange the formula to solve for the rate of change of current (dI/dt).
dI/dt = -ε / L
That's how you can calculate the rate at which current must change through the solenoid using the concept of induced emf.