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 exists only when the electric field changes with time. If the electric field is constant or depends only on position, displacement current is zero.

Answer 1

The Correct Option is A

Step 1: Understanding displacement current.
Displacement current is a concept introduced by Maxwell and is associated with a time-varying electric field. The displacement current density is given by \[ J_d = \varepsilon_0 \frac{\partial E}{\partial t} \] Thus, displacement current exists only when the electric field changes with time.
Step 2: Examine Region I.
In Region I, \[ E_x = E_0 \sin (kz - \omega t) \] This electric field depends on both position $z$ and time $t$. Since it contains the term $\omega t$, the electric field varies with time. Therefore, \[ \frac{\partial E}{\partial t} \neq 0 \] Hence displacement current exists in this region.
Step 3: Examine the remaining regions.
Region II: \[ E_x = E_0 \] This is constant and does not vary with time, so \[ \frac{\partial E}{\partial t} = 0 \] Region III: \[ E_x = E_0 \sin kz \] This depends only on position $z$ and not on time. Region IV: \[ E_x = E_0 \cos kz \] Again, this depends only on position and not on time. Therefore, in Regions II, III and IV the electric field does not change with time, so no displacement current exists.
Step 4: Conclusion.
Displacement current exists only where the electric field varies with time, which occurs in Region I.
Final Answer: I.