This question involves a system of two coils where one coil is inside the other. Let's break down the problem:

Given Data:

• Inner coil (black):
  • Number of turns: $N_1 = 4500$
  • Length: $l_1 = 30 \, \text{cm}$
  • Radius: $r_1 = 5 \, \text{cm}$
• Outer coil (blue):
  • Number of turns: $N_2 = 70$
  • Length: $l_2 = 30 \, \text{cm}$
• Current in outer coil: $I_2 = 0.31 \, \text{A}$
• Rate of change of current: $\frac{dI_2}{dt} = 3.0 \times 10^3 \, \text{A/s}$

Questions:

1. Magnetic field through each turn of the inner coil: This can be calculated by finding the magnetic field produced by the outer coil at the location of the inner coil. Using Ampère's law and the formula for the magnetic field inside a solenoid, we get the magnetic field $B_2$ for the outer coil.

   $B = \mu_0 \frac{N_2}{l_2} I_2$

2. Induced EMF in the inner coil due to changing current in the outer coil: We will calculate the induced electromotive force (EMF) in the inner coil using Faraday’s law of induction, given the change in current in the outer coil.

   $\mathcal{E}_1 = - N_1 \frac{d\Phi}{dt}$ where $\Phi$ is the magnetic flux through the inner coil. We can calculate the flux using the magnetic field $B_2$ and the area of the inner coil’s cross section.

3. Electromagnetic force (Lorentz force) in the inner coil: The force on the current-carrying wire in the inner coil can be found using the Lorentz force law:

   $F = I_1 L B$

   where $L$ is the length of the inner coil, $I_1$ is the current in the inner coil (induced by the changing magnetic field), and $B$ is the magnetic field.

Let me know if you'd like to proceed with calculating these values step-by-step or if you have any specific questions about these steps!