Step 1: The electric flux through a surface is defined by the formula \( \Phi_E = \vec{E} \cdot \vec{A} \). The area vector \( \vec{A} \) is expressed as \( A \hat{n} \). Given a square surface with a side length of 1 cm, the area is calculated as \( A = (1 \, \text{cm})^2 = 1 \times 10^{-4} \, \text{m}^2 \).
Step 2: The electric flux can also be expressed as \( \Phi_E = E A \cos \theta \), where \( E = 100 \, \text{N/C} \) is the magnitude of the electric field, \( A = 1 \times 10^{-4} \, \text{m}^2 \) is the area of the surface, and \( \theta \) is the angle between the electric field vector and the surface's normal vector.
Step 3: The value of \( \cos \theta \) is determined using the dot product of \( \vec{E} \) and \( \hat{n} \). The formula for \( \cos \theta \) is \( \cos \theta = \frac{\vec{E} \cdot \hat{n}}{|\vec{E}|} \). Given \( \vec{E} = 100 \, \hat{i} \, \text{N/C} \) and \( \hat{n} = 0.8 \hat{i} + 0.6 \hat{k} \), the dot product is \( \vec{E} \cdot \hat{n} = (100)(0.8) + (0)(0.6) = 80 \). The magnitude of the electric field is \( |\vec{E}| = 100 \, \text{N/C} \). Therefore, \( \cos \theta = \frac{80}{100} = 0.8 \).
Step 4: The electric flux is calculated as \( \Phi_E = 100 \times 1 \times 10^{-4} \times 0.8 = 8 \times 10^{-3} \, \text{N m}^2/\text{C} \).
Consequently, the electric flux through the surface is \( 8 \times 10^{-3} \, \text{N m}^2/\text{C} \).