The magnetic flux φ$\phi$ (in weber) linked with a closed circuit of resistance 8 Ω$8 \, \Omega$ varies with time (in seconds) as φ = 5t² - 36t + 1$\phi = 5t^2 - 36t + 1$. The induced current in the circuit at t = 2 s$t = 2 \text{ s}$ is ________ A.

Related Formula

Core Logic

Assume a current i$i$ flows through the larger square loop of side L$L$. The magnetic field generated by it at its center acts as a uniform field across the very small inner loop of side $\ell$.

The magnetic field at the center of the large square loop (distance d = L/2$d = L/2$ from each side, angles 45^°$45^\circ$):

Step 2: Mutual Inductance Calculation

Flux linkage for the inner loop:

Given L = ²$L = \ell^2$:

Mutual inductance M$M$:

Using μ₀ = 4π × 10⁻⁷$\mu_0 = 4\pi \times 10^{-7}$:

Comparing with √(x) × 10⁻⁷$\sqrt{x} \times 10^{-7}$, we get x = 128$x = 128$.

Chapter Mix

Class 12 Physics: Electromagnetic Induction

Related Formula

Core Logic

Given data: Initial Magnetic Field, Bᵢ = 5000 T$B_i = 5000\mathrm{\,T}$ Final Magnetic Field, Bf = 3000 T$B_f = 3000\mathrm{\,T}$ Time interval, Δ t = 2 s$\Delta t = 2\mathrm{\,s}$ Diameter, d = 0.02 m ⇒ r = 0.01 m$d = 0.02\mathrm{\,m} \Rightarrow r = 0.01\mathrm{\,m}$ Induced emf, ε = 22 V$\varepsilon = 22\mathrm{\,V}$

Change in magnetic field magnitude |Δ B| = 5000 - 3000 = 2000 T$|\Delta B| = 5000 - 3000 = 2000\mathrm{\,T}$. Area of the coil A = π r² = π (0.01)² = 10⁻⁴π m²$A = \pi r^2 = \pi (0.01)^2 = 10^{-4}\pi \mathrm{\,m^2}$.

Step 2: Equation Evaluation

Using Faraday's Law:

22 = N (0.1π)$22 = N (0.1\pi)$ Taking π ≈ 22/7$\pi \approx 22/7$:

Chapter Mix

Class 12 Physics: Electromagnetic Induction