Answer to Question #289740 in Physics for Amina Zaheer

Question #289740

show that unit vector normal to the surface n= ∇φ ÷ | ∇φ | . find n for the ellipsoid.


φ = a(x^2) + b(y^2) +c(z^2)

1
Expert's answer
2022-01-23T16:14:57-0500

Given:

φ=ax2+by2+cz2φ = ax^2 + by^2+cz^2


The gradient of scalar field is given by

φ=φxi^+φyj^+φzk^\nabla \varphi=\frac{\partial \varphi}{\partial x}\hat i+\frac{\partial \varphi}{\partial y}\hat j+\frac{\partial \varphi}{\partial z}\hat k

Hence

φ=2axi^+2byj^+2czk^\nabla \varphi=2ax\hat i+2by\hat j+2cz\hat k

φ=2(ax)2+(by)2+(cz)2|\nabla \varphi|=2\sqrt{(ax)^2 +(by)^2+(cz)^2}

n=φφ=axi^+byj^+czk^(ax)2+(by)2+(cz)2{\vec n}=\frac{\nabla \varphi}{|\nabla \varphi|}=\frac{ax\hat i+by\hat j+cz\hat k}{\sqrt{(ax)^2 +(by)^2+(cz)^2}}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment