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 exists in regions where there is a time-varying electric field. In Region I, the electric field is: \[ E_x = E_0 \sin(kz - \omega t). \] This electric field varies with time due to the presence of the term $-\omega t$. The displacement current density $J_d$ is given by: \[ J_d = \epsilon_0 \frac{\partial E_x}{\partial t}. \] Differentiating $E_x$ with respect to time: \[ \frac{\partial E_x}{\partial t} = -\omega E_0 \cos(kz - \omega t). \] Thus, a displacement current exists in Region I. In the other regions (II, III, IV), the electric field does not vary with time, so there is no displacement current. Therefore, the displacement current exists in: \[ \boxed{\text{Region I}}. \]