1. Electric Flux Calculation

The electric flux $\Phi_E$ through a surface is given by:

$\Phi_E = E \cdot A \cdot \cos(\theta)$

Where:

• $E = 500 \, \text{N/C}$ (electric field magnitude)
• $A = \pi R^2$ (area of the loop)
• $R = 0.2 \, \text{m}$ (radius of the loop)
• $\theta = 30^\circ$ (angle between the electric field and the normal to the loop's surface)

Step 1: Calculate Area

$A = \pi (0.2)^2 = 0.04\pi \, \text{m}^2$

Step 2: Substitute Values

$\Phi_E = 500 \times 0.04\pi \times \cos(30^\circ)$

Since $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$,

$\Phi_E \approx 500 \times 0.04\pi \times 0.866$

$\Phi_E \approx 54.4 \, \text{Nm}^2/\text{C}$

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2. Flux When Loop is Perpendicular

When the loop is rotated such that its plane is perpendicular to the electric field, the angle $\theta$ becomes $0^\circ$, so:

$\Phi_E = E \cdot A \cdot \cos(0^\circ) = E \cdot A$

$\Phi_E = 500 \times 0.04\pi$

$\Phi_E \approx 62.83 \, \text{Nm}^2/\text{C}$

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Final Answer:

1. Electric flux at $30^\circ$: $\Phi_E \approx 54.4 \, \text{Nm}^2/\text{C}$
2. Electric flux when perpendicular: $\Phi_E \approx 62.83 \, \text{Nm}^2/\text{C}$

Would you like a more detailed breakdown of any step?

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Related Questions:

1. How does the electric flux change if the electric field magnitude doubles?
2. What happens if the radius of the loop is halved?
3. How does the flux change if the angle becomes $90^\circ$?
4. How is electric flux affected if the loop is placed in a non-uniform electric field?
5. What physical quantities influence the electric flux through a surface?

Tip: Electric flux depends on the angle between the field and the normal to the surface, not the surface itself!