PHY2602 Electromagnetism Assignment 2 Solutions
Memo
1.
Electric fields are conservative vector fields. Therefore,
⃗ = ⃗0 everywhere.
∇×E
(i)
Given:
⃗ = 8 [xyx̂ + 2yz ŷ + 3zxẑ]
E
we calculate:
x̂ ŷ ẑ
⃗ =
∇×E ∂ ∂ ∂
.
∂x ∂y ∂z
xy 2yz 3zx
Expanding this determinant:
" # " # " #
∇×E ⃗ = 8 ∂(3zx) − ∂(2yz) x̂ + 8 ∂(xy) − ∂(3zx) ŷ + 8 ∂(2yz) − ∂(xy) ẑ.
∂y ∂z ∂z ∂x ∂x ∂y
Simplifying: h i
= 8 − 2zyx̂ − (3z)ŷ + (−x)ẑ ̸= 0.
Hence, this equation does not represent an electric field.
(ii)
Given:
⃗ = 5 y 2 x̂ + (z 2 + xy)ŷ + 2zyẑ ,
E
we calculate:
x̂ ŷ ẑ
⃗ =
∇×E ∂ ∂ ∂
.
∂x ∂y ∂z
2 2
y z + xy 2zy
Expanding this determinant:
" # " # " #
2 2 2 2
⃗ =5 ∂(2zy) ∂(z + xy) ∂(y ) ∂(2zy) ∂(z + xy) ∂(y )
∇×E − x̂ + 5 − ŷ + 5 − ẑ.
∂y ∂z ∂z ∂x ∂x ∂y
1
Memo
1.
Electric fields are conservative vector fields. Therefore,
⃗ = ⃗0 everywhere.
∇×E
(i)
Given:
⃗ = 8 [xyx̂ + 2yz ŷ + 3zxẑ]
E
we calculate:
x̂ ŷ ẑ
⃗ =
∇×E ∂ ∂ ∂
.
∂x ∂y ∂z
xy 2yz 3zx
Expanding this determinant:
" # " # " #
∇×E ⃗ = 8 ∂(3zx) − ∂(2yz) x̂ + 8 ∂(xy) − ∂(3zx) ŷ + 8 ∂(2yz) − ∂(xy) ẑ.
∂y ∂z ∂z ∂x ∂x ∂y
Simplifying: h i
= 8 − 2zyx̂ − (3z)ŷ + (−x)ẑ ̸= 0.
Hence, this equation does not represent an electric field.
(ii)
Given:
⃗ = 5 y 2 x̂ + (z 2 + xy)ŷ + 2zyẑ ,
E
we calculate:
x̂ ŷ ẑ
⃗ =
∇×E ∂ ∂ ∂
.
∂x ∂y ∂z
2 2
y z + xy 2zy
Expanding this determinant:
" # " # " #
2 2 2 2
⃗ =5 ∂(2zy) ∂(z + xy) ∂(y ) ∂(2zy) ∂(z + xy) ∂(y )
∇×E − x̂ + 5 − ŷ + 5 − ẑ.
∂y ∂z ∂z ∂x ∂x ∂y
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