Help with Maxwell stress tensor

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my think

if ## \hat{r} = \sin(θ) \cos( φ) \hat{x} +\sin(θ) \sin( φ) \hat{y} +\cos(θ) \hat{z} ##
## da = R^2 \sin(θ) dθdφ \hat{r} = da_{x} \hat{x} + da_{x} \hat{y} + da_{z} \hat{z}##
So
##da_{x} = R^2 \sin^2(θ) \cos(φ) dθdφ ##
##da_{y} = R^2 \sin^2(θ) \sin(φ) dθdφ ##
##da_{z} = R^2 \sin(θ) \cos(θ) dθdφ ##
where
## \int_{0}^{2π} \cos(φ) \,dφ =\int_{0}^{2π} \sin(φ) \,dφ = 0 ##

##\overleftrightarrow{T} ⋅ da = 0 + 0 + T_{zz} ⋅ da_{z} ##

##T_{zz} ⋅ da_{z} = \frac{ε_{0}}{2} \left(\frac{Q}{4πε_{o}R^2}\right)^2(\cos^2(θ) - \sin^2(θ)) ⋅ R^2\sin(θ) \cos(θ) dθdφ##

But why in text book give by
## \frac{ε_{0}}{2} \left(\frac{Q}{4πε_{o}R^2}\right)^2 ⋅ R^2\sin(θ) \cos(θ) dθdφ = \frac{ε_{0}}{2} \left(\frac{Q}{4πε_{o}R}\right)^2\sin(θ) \cos(θ) dθdφ##

where are ##(\cos^2(θ) - \sin^2(θ))## ?. ##(\cos^2(θ) - \sin^2(θ))## is missing .
I don't understand
 

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  • #2
Orodruin
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The x and y components do not vanish. There is a ##\varphi## dependence also in ##T_{zx}## and ##T_{zy}## that you need to take into account. You cannot just integrate the area element assuming that they are constant.
 
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