This problem involves the relationship between a current-carrying conductor and a loop in its vicinity, examining the induced effects like magnetic field and electromotive force (emf). Here's how we approach the problem:

Part a) Magnetic field at the location of the loop

To find the magnetic field produced by a long straight current-carrying conductor at the location of the loop, we use Ampère's Law. The magnetic field $B$ at a distance $r$ from a long straight conductor carrying a current $i$ is given by:

$B = \frac{\mu_0 i}{2 \pi r}$

Where:

• $B$ is the magnetic field,
• $\mu_0$ is the permeability of free space ($\mu_0 = 4\pi \times 10^{-7} \, \text{T·m/A}$),
• $i$ is the current,
• $r$ is the distance from the wire.

For the loop, you may need to compute the contribution of the magnetic field over the entire loop, integrating along the path. This often requires using the geometry of the situation and may involve dividing the loop into small segments to calculate the field at each point and then summing the contributions.

Part b) Induced electromotive force (emf) in the loop

The induced emf in the loop due to the changing current in the conductor can be calculated using Faraday's Law of Induction:

$\mathcal{E} = - \frac{d\Phi}{dt}$

Where:

• $\mathcal{E}$ is the induced emf,
• $\Phi$ is the magnetic flux through the loop,
• $\frac{d\Phi}{dt}$ is the rate of change of magnetic flux.

Since the magnetic field depends on the current, and the current is changing with time, you can calculate the rate of change of magnetic flux by first determining the magnetic flux through the loop and then differentiating it with respect to time.

Part c) Calculation of the induced emf

Given:

• $a = 5.0 \, \text{cm}$,
• $b = 30 \, \text{cm}$,
• $L = 17 \, \text{cm}$,
• $\frac{di}{dt} = -5 \, \text{A/s}$,

The induced emf will depend on how these parameters influence the rate of change of magnetic flux. We need to find the magnetic flux $\Phi$, which is:

$\Phi = B \times A$

Where $A$ is the area of the loop, and $B$ is the magnetic field at the location of the loop. After calculating the flux, we differentiate it with respect to time to find the induced emf.

Would you like to go into the detailed step-by-step calculations for these parts?