GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 38 of 65

Marks: +1, -0

Let

a_{r}, a_{ϕ,} a_{z}

be unit vector along r, ϕ & z direction respectively in cylindrical coordinate system for the electric flux density given by

D=(15 a r+2 r a ϕ-3 r z a z) C {/} m^{2} .

The total flux in coulomb emanating from the volume enclosed by solid cylinder of radius

3 {m}

and height

5 {m}

oriented along the

z

axis with its base at origin is

Solution:

{D}↖{→}=(15 {a}↖{∧}_{r}+2 r {a}↖{∧}_{φ}-3 r z {a}↖{∧}_{z}) C {/} m^{2}

{Q}_{\;\text "enclosed "\;}=∫ { {D}↖{→} ⋅ { {ds}↖{→}=∫(∇ ⋅ { {D}↖{→}) {dv}

∇ ⋅ {D}↖{→}=\;{1}/{r} \;{∂}/{∂ r} r D_{r}+\;{1}/{r} \;{∂}/{∂ϕ} D_{ϕ}+\;{∂ D_{z}}/{∂ z}

=\;{1}/{r} \;{∂}/{∂ r}(15 r)+\;{1}/{r} \;{∂}/{∂ φ}(2 r)+\;{∂}/{∂ z}(-3 r z)

=\;{15}/{r}+(-3 r)=\;{15}/{r}-3 r

Q_{enclosed}=∫↙{v}(\;{15}/{r}-3 r) r d r d ϕ d z

=∫↙{0}↖{5} ∫↙{0}↖{2 π} ∫↙{0}↖{3}(15-3 r^{2}) d r d ϕ d z

=∫↙{0}↖{5} ∫↙{0}↖{2 π}{|15 r-\;{3 r^{3}}/{3}|}_{0}^{3} d ϕ d z

=∫↙{0}↖{5} ∫↙{0}↖{2 π}(45-27) d ϕ d z

=18 × 5 × 2 π=180 π C

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