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To approach this problem, let’s go through each part carefully:

### Part a: Show that the fields \( E(z,t) = A \sin(kz - \omega t) \mathbf{i} \) and \( B(z,t) = A \sin(kz - \omega t) \mathbf{j} \) satisfy the equation \( \nabla \times \mathbf{B} = C \frac{\partial \mathbf{E}}{\partial t} \), given \( \omega = \frac{k}{C} \).

1. **Calculate \( \frac{\partial \mathbf{E}}{\partial t} \):**
   - Given \( \mathbf{E}(z,t) = A \sin(kz - \omega t) \mathbf{i} \),
   - Taking the time derivative:
     \[
     \frac{\partial \mathbf{E}}{\partial t} = -A \omega \cos(kz - \omega t) \mathbf{i}
     \]

2. **Calculate \( \nabla \times \mathbf{B} \):**
   - Given \( \mathbf{B}(z,t) = A \sin(kz - \omega t) \mathbf{j} \),
   - We need to compute \( \nabla \times \mathbf{B} \).
   - In Cartesian coordinates, the curl of \( \mathbf{B} \) for a function only dependent on \( z \) is:
     \[
     \nabla \times \mathbf{B} = \left( \frac{\partial B_y}{\partial z} \right) \mathbf{i} - \left( \frac{\partial B_x}{\partial z} \right) \mathbf{j}
     \]
   - With \( \mathbf{B} = A

One of Maxwell's equations for electromagnetic waves (also called Ampère's Law) is ∇ × B = C ∂E/∂t, where E is the electric field, B is the magnetic field, and C is a constant. Show that the fields E(z,t) = A sin(kz - ωt)i and B(z,t) = A sin(kz - ωt)j satisfy the equation for constants A, k, ω, provided ω = k/C.

Learn how to show that the fields E(z,t) = A sin(kz - ωt)i and B(z,t) = A sin(kz - ωt)j satisfy Maxwell's equation ∇ × B = C ∂E/∂t. This problem explores key concepts in electromagnetic waves, vector calculus, and Maxwell's equations. Suitable for college-level students studying physics.

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