Concept:
The induced emf in a coil is given by Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the coil. Mathematically, it can be expressed as:

\(\varepsilon = -\frac{{d\Phi}}{{dt}}\)

Where:
- ε is the induced emf
- Φ is the magnetic flux through the coil
- t is the time

The magnetic flux through a coil is given by the product of the magnetic field (B) and the area (A) of the coil. Mathematically, it can be expressed as:

\(\Phi = B \cdot A\)

Since the magnetic field is directly proportional to the current passing through the coil (B ∝ I), we can write:

\(\varepsilon = -\frac{{d(B \cdot A)}}{{dt}} = -B \cdot \frac{{dA}}{{dt}}\) [Using chain rule of differentiation]

Explanation:
Since the current in both coils is increased at the same constant rate, we can assume that the rate of change of current with respect to time is the same for both coils.

Let's consider the induced emf in the first coil (Coil 1) with self-inductance 8 mH. We can write:

\(\varepsilon_1 = -B_1 \cdot \frac{{dA_1}}{{dt}}\) ...(1)

Similarly, for the second coil (Coil 2) with self-inductance 2 mH, we can write:

\(\varepsilon_2 = -B_2 \cdot \frac{{dA_2}}{{dt}}\) ...(2)

Since the rate of change of current with respect to time is the same for both coils, we can write:

\(\frac{{dA_1}}{{dt}} = \frac{{dA_2}}{{dt}}\) ...(3)

Now, let's compare the magnetic fields (B1 and B2) in both coils. The magnetic field is directly proportional to the current passing through the coil. Since the rate of change of current is the same for both coils, we can assume that the currents in both coils are in the same ratio as their self-inductances.

Let the current in Coil 1 be I1 and the current in Coil 2 be I2. We have:

\(\frac{{I_1}}{{I_2}} = \frac{{\text{{Self-inductance of Coil 1}}}}{{\text{{Self-inductance of Coil 2}}}} = \frac{{8 \, \text{{mH}}}}{{2 \, \text{{mH}}}} = 4\) ...(4)

Therefore, we can write:

\(I_1 = 4I_2\) ...(5)

Since the magnetic field is directly proportional to the current, we can write:

\(B_1 = kI_1\) ...(6)

\(B_2 = kI_2\) ...(7)

Where k is a constant of proportionality.

Substituting equations (5), (6), and (7) into equations (1) and (2), we get:

\(\varepsilon_1 = -kI_1 \cdot \frac