The functions V 1 (, , z) and V 2 (, , z) both satisfy Laplaces equation in the region a b, 0 v 2, L z L each is zero on the surfaces b for L z L z L for a b and z L for a b and each is 100 V on the surface a for L z L (a) In the region specified, is Laplaces equation satisfied by the functions V 1 V 2 , V 1 V 2 , V 1 3, and V 1 V 2 (b) On the boundary surfaces specified, are the potential values given in this problem obtained from the functions V 1 V 2 , V 1 V 2 , V 1 3, and V 1 V 2 (c) Are the functions V 1 V 2 , V 1 V 2 , V 1 3, and V 1 V 2 identical with V 1

Question: The functions V 1 (, , z) and V 2 (, , z) both satisfy Laplaces equation in the region a < < b,

The functions V₁(ρ, ϕ, z) and V₂(ρ, ϕ, z) both satisfy Laplace’s equation in the region a < ρ < b, 0 ≤ v < 2π, −L < z < L; each is zero on the surfaces ρ = b for −L < z < L; z = −L for a < ρ < b; and z = L for a < ρ < b; and each is 100 V on the surface ρ = a for −L < z < L.

(a) In the region specified, is Laplace’s equation satisfied by the functions V₁ + V₂, V₁ − V₂, V₁ + 3, and V₁V₂?

(b) On the boundary surfaces specified, are the potential values given in this problem obtained from the functions V₁ + V₂, V₁ − V₂, V₁ + 3, and V₁V₂?

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