Question

A plane surface, in the shape of a square of side 1 cm, is placed in an electric field \( \vec{E} = (100 \, \text{N/C}) \hat{i \) such that the unit vector normal to the surface is given by \( \hat{n} = 0.8 \hat{i} + 0.6 \hat{k} \). Find the electric flux through the surface.}

Hint:

The electric flux depends on the angle between the electric field and the normal to the surface. For maximum flux, the field is normal to the surface.

Answer 1

Step 1: The electric flux through the surface is given by the formula:
\[ \Phi_E = \vec{E} \cdot \vec{A} \] where \( \vec{A} = A \hat{n} \) is the area vector. Since the surface is square with side length 1 cm, the area is:
\[ A = (1 \, \text{cm})^2 = 1 \times 10^{-4} \, \text{m}^2 \] Step 2: The electric flux is:
\[ \Phi_E = E A \cos \theta \] where \( E = 100 \, \text{N/C} \), \( A = 1 \times 10^{-4} \, \text{m}^2 \), and \( \theta \) is the angle between the electric field vector and the unit normal vector to the surface.
Step 3: To find \( \cos \theta \), we use the dot product of \( \vec{E} \) and \( \hat{n} \):
\[ \cos \theta = \frac{\vec{E} \cdot \hat{n}}{|\vec{E}|} \] \[ \vec{E} = 100 \, \hat{i} \, \text{N/C}, \quad \hat{n} = 0.8 \hat{i} + 0.6 \hat{k} \] \[ \vec{E} \cdot \hat{n} = (100)(0.8) + (0)(0.6) = 80 \] \[ |\vec{E}| = 100 \, \text{N/C} \] \[ \cos \theta = \frac{80}{100} = 0.8 \] Step 4: Now, calculate the electric flux:
\[ \Phi_E = 100 \times 1 \times 10^{-4} \times 0.8 = 8 \times 10^{-3} \, \text{N m}^2/\text{C} \] Thus, the electric flux through the surface is \( 8 \times 10^{-3} \, \text{N m}^2/\text{C} \).