Step 1: Check ∇ · 𝐁 = 0

The magnetic field is given as: 𝐁 = B₀ sin(πx/L) cos(πy/L) e^−ωt 𝑘̂
This field has only a z-component: Bz(x, y, t).
Therefore,

∇ · 𝐁 = ∂Bx/∂x + ∂By/∂y + ∂Bz/∂z = 0 + 0 + 0 = 0

✅ The divergence of B is zero, as required by Maxwell’s equation.

Step 2: Apply Faraday’s Law

Faraday’s law states: ∇ × 𝐄 = −∂𝐁/∂t

Compute the time derivative of 𝐁:
∂𝐁/∂t = ∂/∂t [B₀ sin(πx/L) cos(πy/L) e^−ωt] 𝑘̂
= −ωB₀ sin(πx/L) cos(πy/L) e^−ωt 𝑘̂
Therefore: ∇ × 𝐄 = ωB₀ sin(πx/L) cos(πy/L) e^−ωt 𝑘̂

Step 3: Choose Electric Field with Components in x-y Plane

Assume 𝐄(x, y, t) = E_x î + E_y ĵ and E_z = 0

Then:

(∇ × 𝐄)_z = ∂E_y/∂x − ∂E_x/∂y
To satisfy Faraday’s law, we set: ∂E_y/∂x − ∂E_x/∂y = ωB₀ sin(πx/L) cos(πy/L) e^−ωt

Step 4: Choose E_x = 0 to Simplify

This gives:

∂E_y/∂x = ωB₀ sin(πx/L) cos(πy/L) e^−ωt
Now integrate w.r.t x:

E_y(x, y, t) = ∫ ωB₀ sin(πx/L) cos(πy/L) e^−ωt dx + g(y,t)

= −(ωB₀L/π) cos(πx/L) cos(πy/L) e^−ωt + g(y,t)

Assume g(y,t) = 0 for simplicity.

Step 5: Verify Faraday’s Law

We have:
∂E_y/∂x = ωB₀ sin(πx/L) cos(πy/L) e^−ωt
∂E_x/∂y = 0
So: ∇ × 𝐄 = ωB₀ sin(πx/L) cos(πy/L) e^−ωt 𝑘̂ = −∂𝐁/∂t

✅ Thus, Faraday’s law is fully satisfied by this electric field.