Answer to Question #185023 in Physics for Satyam

Question #185023

Express the field D = (x2 + y2)−1(xax + yay) in cylindrical components and cylindrical

variables.


1
Expert's answer
2021-04-26T17:13:03-0400
x=ρcosφ,y=ρsinφD=1ρ(cosφax+sinφay)x = ρ \cos φ, y = ρ \sin φ\\\bold{D}=\frac{1}{\rho}(\cos φ\bold{a_x}+ \sin φ\bold{a_y})

Then


Dρ=1ρ(cosφaxaρ+sinφayaρ)=1ρ(cos2φ+sin2φ)=1ρD_\rho=\frac{1}{\rho}(\cos φ\bold{a_x\cdot a_\rho}+ \sin φ\bold{a_y\cdot a_\rho})\\=\frac{1}{\rho}(\cos^2 φ+ \sin^2 φ)=\frac{1}{\rho}

Dϕ=1ρ(cosφaxaϕ+sinφayaϕ)=1ρ(sinφcosφ+sinφcosφ)=0D_\phi=\frac{1}{\rho}(\cos φ\bold{a_x\cdot a_\phi}+ \sin φ\bold{a_y\cdot a_\phi})\\=\frac{1}{\rho}(-\sin φ\cos φ+ \sin φ\cos φ)=0

So,


D=1ρaρ\bold{D}=\frac{1}{\rho}\bold{a_\rho}


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