Question

In the four regions, I, II, III, and IV, the electric fields are described as: Region I: $E_x = E_0 \sin(kz - \omega t)$
Region II: $E_x = E_0$
Region III: $E_x = E_0 \sin kz$
Region IV: $E_x = E_0 \cos kz$
The displacement current will exist in the region:

• A. I
• B. IV
• C. II
• D. III

Hint:

Displacement current arises from time-varying electric fields. Check for terms involving $t$ in the electric field expressions to identify regions with displacement current.

Answer 1

The Correct Option is A

Displacement current is present where the electric field is time-dependent. In Region I, the electric field is given by \(E_x = E_0 \sin(kz - \omega t)\). The term \(-\omega t\) indicates this field's time variation. The displacement current density \(J_d\) is defined as \(J_d = \epsilon_0 \frac{\partial E_x}{\partial t}\). Differentiating \(E_x\) with respect to time yields \(\frac{\partial E_x}{\partial t} = -\omega E_0 \cos(kz - \omega t)\). Consequently, a displacement current exists in Region I. In Regions II, III, and IV, the electric field is static, meaning no displacement current is present. Therefore, displacement current is found in: \[\boxed{\text{Region I}}.\]