### Related Formula According to Faraday's Law of Electromagnetic Induction, the magnitude of induced EMF (e$e$) is: e = fracDelta phiDelta t$e = \frac{\Delta \phi}{\Delta t}$ where magnetic flux phi$\phi$ is defined via dot product: phi = vecB cdot vecA = B A cos theta$\phi = \vec{B} \cdot \vec{A} = B A \cos \theta$ ### Core Logic Let the East-West plane lie vertical along the x-z plane. The normal vector to the loop points along the North direction (along hatj$\hat{j}$): vecA = (0.1 mathrm~m)^2 hatj = 0.01 hatj mathrm~m^2$\vec{A} = (0.1 \mathrm{~m})^2 \hat{j} = 0.01 \hat{j} \mathrm{~m^2}$ The uniform magnetic field points North-East, meaning it is at an angle of 45^circ$45^\circ$ to the North vector direction: vecB = B cos 45^circ hati + B sin 45^circ hatj = frac0.2sqrt2hati + frac0.2sqrt2hatj$\vec{B} = B \cos 45^\circ \hat{i} + B \sin 45^\circ \hat{j} = \frac{0.2}{\sqrt{2}}\hat{i} + \frac{0.2}{\sqrt{2}}\hat{j}$

### Step 1: Calculate Initial Flux $phi_i = vecB cdot vecA = left( frac0.2sqrt2 right) times 0.01 = frac2 times 10^-3sqrt2 = sqrt2 times 10^-3 mathrm~Wb$$\phi_i = \vec{B} \cdot \vec{A} = \left( \frac{0.2}{\sqrt{2}} \right) \times 0.01 = \frac{2 \times 10^{-3}}{\sqrt{2}} = \sqrt{2} \times 10^{-3} \mathrm{~Wb}$ ### Step 2: Find Induced EMF Since the field goes steadily to zero in $ ### Step 2: Find Induced EMF Since the field goes steadily to zero in $\Delta t = 1 \mathrm{~s}, the final flux value $, the final flux value $\phi_f = 0: $: $e = frac|0 - phi_i|1 = sqrt2 times 10^-3 mathrm~V$e = \frac{|0 - \phi_i|}{1} = \sqrt{2} \times 10^{-3} \mathrm{~V}$ ### Step 3: Extract x Comparing this with the target expression $ ### Step 3: Extract x Comparing this with the target expression $\sqrt{x} \times 10^{-3} \mathrm{~V}: $: $x = 2 ### Pattern Recognition Be careful when identifying angles between directional plane descriptors. An East-West plane has a normal axis directed North-South. A North-East field makes a clean $ ### Pattern Recognition Be careful when identifying angles between directional plane descriptors. An East-West plane has a normal axis directed North-South. A North-East field makes a clean $45^\circ$ angle relative to this structural normal line. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Electromagnetic Induction