The current flowing in a solenoid that is 20.9 cm long and has a radius of 2.40 cm and 552 turns decreases from 3.51 A to 1.03 A in 0.250 s. Determine the magnitude of the induced electric field inside the solenoid 1.20 cm from its center.

To determine the magnitude of the induced electric field inside the solenoid, we can use Faraday's Law of electromagnetic induction. According to Faraday's Law, the magnitude of the induced electric field is given by:

|E| = - (dφ/dt) / A

Where:
|E| is the magnitude of the induced electric field.
dφ/dt is the rate of change of the magnetic flux through the area A.
A is the cross-sectional area of the solenoid.

To calculate the rate of change of the magnetic flux (dφ/dt), we need to first calculate the change in magnetic flux (∆φ) during the given time interval (∆t).

The magnetic flux (∆φ) can be calculated using the equation:

∆φ = B * A

Where:
B is the magnetic field inside the solenoid.
A is the cross-sectional area of the solenoid.

To find the magnetic field (B), we can use Ampere's Law, which states that the magnetic field inside a solenoid is given by:

B = μ₀ * n * I

Where:
μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A).
n is the number of turns per unit length (turns/m).
I is the current flowing through the solenoid (A).

Given:
Length of the solenoid, L = 20.9 cm = 0.209 m
Radius of the solenoid, r = 2.40 cm = 0.024 m
Number of turns, N = 552
Initial current, I₁ = 3.51 A
Final current, I₂ = 1.03 A
Time interval, ∆t = 0.250 s
Position from the center, r' = 1.20 cm = 0.012 m

Step 1: Calculate the cross-sectional area of the solenoid.
The cross-sectional area of a solenoid is given by the formula:

A = π * r²

Substituting the values, we get:
A = 3.1416 * (0.024 m)² = 0.0045233 m²

Step 2: Calculate the magnetic field inside the solenoid (B) at the initial and final current values.
Using Ampere's Law:

B₁ = μ₀ * n * I₁
B₂ = μ₀ * n * I₂

Where:
μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)

Substituting the values, we get:
B₁ = 4π × 10⁻⁷ T·m/A * (552 turns/m) * (3.51 A)
B₂ = 4π × 10⁻⁷ T·m/A * (552 turns/m) * (1.03 A)

Step 3: Calculate the change in magnetic flux (∆φ) during the given time interval.
The change in magnetic flux (∆φ) can be given by:

∆φ = B₂ * A - B₁ * A

Substituting the values, we get:
∆φ = (B₂ - B₁) * A

Step 4: Calculate the rate of change of the magnetic flux (dφ/dt).
The rate of change of the magnetic flux (dφ/dt) can be calculated by dividing the change in magnetic flux (∆φ) by the given time interval (∆t).

(dφ/dt) = (∆φ) / ∆t

Substituting the values, we get:
(dφ/dt) = (∆φ) / 0.250 s

Step 5: Calculate the magnitude of the induced electric field inside the solenoid at the given position (r').

|E| = - (dφ/dt) / A

Substituting the calculated value of (dφ/dt) and the cross-sectional area A, we can determine the magnitude of the induced electric field.