329sp09hw5sol - ECE-329 Spring 2009 Homework 5 — Solution...

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Unformatted text preview: ECE-329 Spring 2009 Homework 5 — Solution February 17, 2009 1. Verifying vector calculus identities, ∇ × ( ∇ Φ) = and ∇ · ( ∇ × A ) = 0 . a) The gradient of a scalar field Φ is defined as ∇ Φ = ∂ Φ ∂x ˆ x + ∂ Φ ∂y ˆ y + ∂ Φ ∂z ˆ z. Taking the curl, we obtain ∇ × ( ∇ Φ) = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z ∂ Φ ∂x ∂ Φ ∂y ∂ Φ ∂z = ∂ ∂y ∂ Φ ∂z- ∂ ∂z ∂ Φ ∂y ˆ x + ∂ ∂z ∂ Φ ∂x- ∂ ∂x ∂ Φ ∂z ˆ y + ∂ ∂x ∂ Φ ∂y- ∂ ∂y ∂ Φ ∂x ˆ z = . b) The curl of a vector field A = A x ˆ x + A y ˆ y + A z ˆ z is defined as ∇ × A = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z A x A y A z = ∂A z ∂y- ∂A y ∂z ˆ x + ∂A x ∂z- ∂A z ∂x ˆ y + ∂A y ∂x- ∂A x ∂y ˆ z. Taking the divergence, we obtain ∇ · ( ∇ × A ) = ∂ ∂x ∂A z ∂y- ∂A y ∂z + ∂ ∂y ∂A x ∂z- ∂A z ∂x + ∂ ∂z ∂A y ∂x- ∂A x ∂y = ∂ 2 A z ∂x∂y- ∂ 2 A y ∂x∂z + ∂ 2 A x ∂y∂z- ∂ 2 A z ∂y∂x + ∂ 2 A y ∂z∂x- ∂ 2 A x ∂z∂y = 0 . 2. Let us verify if A = ( x- y )ˆ x + ( x + y )ˆ y satisfies the following vector identity ∇ × ( ∇ × A ) = ∇ ( ∇ · A )- ∇ 2 A . The left-hand side gives ∇ × ( ∇ × A ) = ∇ × ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z x- y x + y = ∇ × (2ˆ z ) = , 1 ECE-329 Spring 2009 and the right hand side gives ∇ ( ∇ · A )- ∇ 2 A = ∇ ( ∇ · (( x- y )ˆ x + ( x + y )ˆ y ))- ∇ 2 (( x- y )ˆ x + ( x + y )ˆ y ) = ∇ (2)- = , thus, the identity is verified. 3. The charge distribution associated with a point charge Q = 4 π o C can be expressed as ρ ( x,y,z ) = Qδ ( x ) δ ( y ) δ ( z ) where δ ( · ) is the delta function. a) Calculating the closed volume integral ¸ V ρdV. i. If V contains the origin, we have ˛ V ρdV = ˛ V Qδ ( x ) δ ( y ) δ ( z ) dxdydz = Q. ii. If V excludes the origin, we simply have ˛ V ρdV = 0 . b) Above, we found that no matter what the shape or size of the volume V are, as long as V contains the charge distribution ρ, the charge inside the volume is Q. Notice, that we would have found the same, if we were considered instead a point charge Q placed at the origin. Thus, we can argue that the charge density function ρ ( x,y,z ) = Qδ ( x ) δ ( y ) δ ( z ) is a valid description of a point charge.point charge....
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This note was uploaded on 11/16/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

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329sp09hw5sol - ECE-329 Spring 2009 Homework 5 — Solution...

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