If the electric field is given by `(6 hat(i) + 4 hat(j) + 4 hat(k))`, calculate the electric flux through a surface of area 20 units lying in `Y-Z` plane.

Explanation

To calculate the electric flux through a surface in the Y-Z plane given the electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Electric Field and Surface Area**:
The electric field is given as:
\[ \vec{E} = 6 \hat{i} + 4 \hat{j} + 4 \hat{k} \]
The surface area is given as 20 units lying in the Y-Z plane.

2. **Determine the Normal Vector of the Surface**:
Since the surface lies in the Y-Z plane, the normal vector to this surface is along the X-axis. Thus, the unit normal vector can be represented as:
\[ \hat{n} = \hat{i} \]

3. **Calculate the Electric Flux**:
The electric flux (\(\Phi\)) through a surface is given by the dot product of the electric field vector and the area vector:
\[ \Phi = \vec{E} \cdot \vec{A} \]
where \(\vec{A} = A \hat{n}\) and \(A\) is the area of the surface. Here, the area \(A = 20\) units, so:
\[ \vec{A} = 20 \hat{i} \]

4. **Compute the Dot Product**:
Now, we compute the dot product:
\[ \Phi = \vec{E} \cdot \vec{A} = (6 \hat{i} + 4 \hat{j} + 4 \hat{k}) \cdot (20 \hat{i}) \]
This simplifies to:
\[ \Phi = 6 \cdot 20 + 4 \cdot 0 + 4 \cdot 0 = 120 \]

5. **State the Final Result**:
Thus, the electric flux through the surface is:
\[ \Phi = 120 \, \text{N m}^2/\text{C} \]