1. Given data

• Coil $C_1$:

  • Number of turns $N_1 = 500$
  • Radius $r_1 = 1\text{ cm} = 0.01\text{ m}$

• Coil $C_2$:

  • Number of turns $N_2 = 200$
  • Radius $r_2 = 20\text{ cm} = 0.20\text{ m}$
  • Current: $I(t) = 5t^2 - 2t + 3$

We need the induced emf in $C_1$ at $t=1\text{ s}$.

2. Magnetic field due to coil $C_2$ at the center

Since the smaller coil $C_1$ is concentric with the larger coil $C_2$, and $r_1 \ll r_2$, the magnetic field produced by $C_2$ over the area of $C_1$ may be taken approximately uniform and equal to the field at the center of $C_2$.

Magnetic field at the center of a circular coil with $N_2$ turns is

$B = \frac{\mu_0 N_2 I}{2r_2}$

So,

$B(t) = \frac{\mu_0 N_2}{2r_2} I(t)$

3. Flux linked with coil $C_1$

Area of each turn of $C_1$:

$A_1 = \pi r_1^2 = \pi (0.01)^2 = \pi \times 10^{-4}$

Flux through one turn of $C_1$:

$\phi = BA_1 = \frac{\mu_0 N_2 I}{2r_2} A_1$

Total flux linkage with $N_1$ turns:

$\Phi = N_1 \phi = N_1 \frac{\mu_0 N_2 I}{2r_2} A_1$

Induced emf magnitude:

e = \left|\frac{d\Phi}{dt}\right| = N_1 A_1 \frac\mu_0 N_2{2r_2} \left|\frac{dI}{dt}\right|

4. Differentiate the current

$I(t) = 5t^2 - 2t + 3$

$\frac{dI}{dt} = 10t - 2$

At $t=1$ s,

$\left.\frac{dI}{dt}\right|_{t=1} = 10(1)-2 = 8\text{ A/s}$

5. Substitute all values

$e = 500 \cdot (\pi \times 10^{-4}) \cdot \frac{(4\pi \times 10^{-7})\cdot 200}{2\cdot 0.20} \cdot 8$

Now simplify the middle factor:

$\frac{(4\pi \times 10^{-7})\cdot 200}{0.40}$

Since $200/0.40 = 500$,

$= 4\pi \times 10^{-7} \times 500 = 2\pi \times 10^{-4}$

Therefore,

$e = 500 \cdot (\pi \times 10^{-4}) \cdot (2\pi \times 10^{-4}) \cdot 8$

$e = 500 \cdot 16\pi^2 \times 10^{-8}$

$e = 8000\pi^2 \times 10^{-8}$

$e = 8\pi^2 \times 10^{-5}\text{ V}$

Using $\pi^2 \approx 9.87$,

$e \approx 8 \times 9.87 \times 10^{-5} = 78.96 \times 10^{-5}\text{ V}$

$e \approx 7.896 \times 10^{-4}\text{ V}$

In mV,

$e \approx 0.7896\text{ mV}$

6. Match with the form $\dfrac{4}{x}$ mV

Given,

$\frac{4}{x} = 0.7896$

So,

$x = \frac{4}{0.7896} \approx 5.07$

Hence,

$x = 5$

7. Comparison with stored answer

Derived answer: $5$

Stored correct answer: $5$

They agree.