A linearly polarised electromagnetic wave given as `E=E_(0)haticos(kz-omegat)` is incident normally on a perfectly reflecting wall `z=a`. Assuming that the material of the optically inactive, the reflected wave will be give as

Options

A. `E_(r) =E_(0) hati (kz - omegat)`
B. `E_(r)=E_(0) hati "cos" (kz +omegat)`
C. `E_(r) =- E_(0) hati " cos " (kz +omegat)`
D. `E_(r) =E_(0) hati " sin "(kz - omegat)`

Explanation

To find the equation of the reflected wave when a linearly polarized electromagnetic wave is incident normally on a perfectly reflecting wall, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Incident Wave**:
The incident wave is given by:
\[ E = E_0 \hat{i} \cos(kz - \omega t) \]
Here, \(E_0\) is the amplitude, \(\hat{i}\) indicates the direction of polarization, \(k\) is the wave number, \(z\) is the position, \(\omega\) is the angular frequency, and \(t\) is time.

2. **Understanding Reflection**:
When the wave hits a perfectly reflecting wall at \(z = a\), it reflects back. For a perfect reflection, the phase of the reflected wave changes by \(180^\circ\) (or \(\pi\) radians).

3. **Change in Phase**:
The reflected wave will have a phase shift of \(\pi\). Therefore, we can express the reflected wave as:
\[ E_r = -E_0 \hat{i} \cos(kz - \omega t + \pi) \]
The negative sign indicates the phase shift.

4. **Substituting for Reflection**:
Since the wave is reflecting, we also need to consider the direction of \(z\). The wave is now moving in the negative \(z\) direction after reflection. Thus, we replace \(z\) with \(-z\):
\[ E_r = -E_0 \hat{i} \cos(k(-z) - \omega t + \pi) \]

5. **Simplifying the Expression**:
We can simplify this expression:
\[ E_r = -E_0 \hat{i} \cos(-kz - \omega t + \pi) \]
Using the property of cosine, \(\cos(-x) = \cos(x)\), we have:
\[ E_r = -E_0 \hat{i} \cos(kz + \omega t + \pi) \]

6. **Final Expression**:
The cosine function has the property that \(\cos(x + \pi) = -\cos(x)\). Therefore, we can further simplify:
\[ E_r = E_0 \hat{i} \cos(kz + \omega t) \]

### Conclusion:
The equation for the reflected wave is:
\[ E_r = E_0 \hat{i} \cos(kz + \omega t) \]