# Is this vector calculus problem correct?

• Nov 14th 2013, 11:18 AM
mark090480
Is this vector calculus problem correct?
A scalar field $\displaystyle \phi$ is a function of x,z and t only. Vectors $\displaystyle {\bf{E}}$ and $\displaystyle {\bf{H}}$ are defined by:

$\displaystyle {\bf{E}} = \frac{1}{\varepsilon }\left( {\frac{{\partial \phi }}{{\partial {z^{}}}}{\bf{i}} - \frac{{\partial \phi }}{{\partial x}}{\bf{k}}} \right)$

$\displaystyle {\bf{H}} = - \frac{{\partial \phi }}{{\partial t}}{\bf{j}}$

The divergence of a vector field

$\displaystyle {\bf{F}} = {F_1}{\bf{i}} + {F_2}{\bf{j}} + {F_3}{\bf{k}}$

is a scalar given by:

$\displaystyle \nabla \cdot {\bf{F}} = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial t}} + \frac{{\partial {F_3}}}{{\partial z}}$

Show that

$\displaystyle \nabla \cdot {\bf{H}} = 0$

$\displaystyle \nabla \cdot {\bf{H}} = \left( {{\bf{i}}\frac{\partial }{{\partial x}} + {\bf{j}}\frac{\partial }{{\partial t}} + {\bf{k}}\frac{\partial }{{\partial z}}} \right) \cdot \left( { - \frac{{\partial \phi }}{{\partial {t^{}}}}{\bf{j}}} \right)$

$\displaystyle \frac{{\partial {F_2}}}{{\partial t}} = 0$

And show that $\displaystyle \nabla \cdot {\bf{E}} = 0$

$\displaystyle {\bf{E}} = \frac{1}{\varepsilon }\frac{{\partial \phi }}{{\partial {z^{}}}}{\bf{i}} - \frac{1}{\varepsilon }\frac{{\partial \phi }}{{\partial x}}{\bf{k}}$

$\displaystyle \nabla \cdot {\bf{E}} = \left( {{\bf{i}}\frac{\partial }{{\partial x}} + {\bf{j}}\frac{\partial }{{\partial z}} + {\bf{k}}\frac{\partial }{{\partial t}}} \right) \cdot \left( {\frac{1}{\varepsilon }\frac{{\partial \phi }}{{\partial {z^{}}}}{\bf{i}} - \frac{1}{\varepsilon }\frac{{\partial \phi }}{{\partial x}}{\bf{k}}} \right)$

So that gives

$\displaystyle \frac{{\partial {F_1}}}{{\partial x}} = \frac{1}{\varepsilon }\frac{{\partial \phi }}{{\partial {z^{}}}}$

$\displaystyle \frac{{\partial {F_3}}}{{\partial x}} = - \frac{1}{\varepsilon }\frac{{\partial \phi }}{{\partial x}}$

Which will cancel each other out

Given that

$\displaystyle \nabla \times {\bf{E}} = - \mu \frac{{\partial {\bf{H}}}}{{\partial t}}$

where u is a constant show that $\displaystyle \phi$ satisfies the partial differential equation

$\displaystyle \frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} = \mu \varepsilon \frac{{{\partial ^2}\phi }}{{\partial {t^2}}}$

Thanks for feedback.
• Nov 14th 2013, 01:33 PM
chiro
Re: Is this vector calculus problem correct?
Hey mark090480.

The first thing you should do is setup a matrix and calculate its determinant to calculate curl(E) and calculate the LHS accordingly. Based on this and the other constraint you will have two pieces of information which you can combine.
• Nov 14th 2013, 08:58 PM
mark090480
Re: Is this vector calculus problem correct?
Hello Chiro,

Do you mean to solve that last part I should use the curl? The problem I have is that this is all the info I have, I cannot calculate the det as I have no real numbers to use. What about the other parts, are they looking good?

Thanks for any help.
• Nov 15th 2013, 01:03 AM
chiro
Re: Is this vector calculus problem correct?
You can do it symbolically just like you did the del(E). Just set up the matrix symbolically and then use algebra to calculate the determinant.