27: Problem 27.12

INTRO:
A 6.00 cm x 4.20 cm rectangle lies in the xy-plane

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PART A:

What is the electric flux through the rectangle is E = (120 i^ + 100 k^) N/C?

SOLUTION:
To begin, convert the dimensions to meters so that the problem is in SI units.

x = 0.060 m & y = 0.042 m

The area... A_xy = (0.042 m)(0.060 m) → A_xy = 0.00252 m²

Normal vector to the xy-plane: N_v = k^

∴ A = A_xy⋅N_v →A = (0.00252 m²)k^

Some necessary knowledge needed to solve this problem is the rules for the dot product of unit vectors. 

i^⋅k^ = 0
j^⋅k^ = 0
k^⋅k^ = 1

if you need more of a refresher, HERE

Now, using Gauss's law... 
Φ_E = ∫E⋅dA
→ Φ_E = [(120i^+100k^)N/C]⋅[(0.00252 m²)k^]

Φ_E = 0.252 (N⋅m²)/C

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PART B:

What is the electric flux through the rectangle is E = (120 i^ + 100 j^) N/C?

SOLUTION:

Done the same way as in part A...
→ Φ_E = [(120i^+100j^)N/C]⋅[(0.00252 m²)k^]

Φ_E = 0 (N⋅m²)/C