Finding Electric Field in Spherical Electrostatics: Vector Specification?

[Swapnil]

I have a problem in electrostatics in which there is a uniformly charges semi-sphere of radius a with its base in the xy-plane and I want to find the electricfiled at some point h on the z-axis. What I am having trouble with is that how do you specify the vector that goes from an infinitesimal point on the surface of the sphere to the point h on the z axis? Would it be just $$h\hat{r} - a\hat{r}$$?

 

[quasar987]

Vector addition. You want the vector that goes from a point on the sphere to a point (0,0,h) on the z axis. Well, that's $-a\hat{r}+h\hat{z}$.

 

[Swapnil]

Is that valid? Can you mix spherical and rectangular coordinates together like that?

Anyways, how would you find its magnitude? Would you just take the squareroot of the dot product? Something like this:

$$\vec{R} = h\hat{z} - a\hat{r}$$

$$|\vec{R}|^2 = \vec{R}\cdot\vec{R}$$

$$= ( h\hat{z} - a\hat{r} )\cdot( h\hat{z} - a\hat{r} ) = h^2\hat{z}\cdot\hat{z} + a^2\hat{r}\cdot\hat{r} = h^2 + a^2$$

Is this correct? Doesn't look right though...

 

[Dr Transport]

Remember that $$\hat{r} = (a\hat{i} + b\hat{j}+c\hat{k})$$ so you can take the dot product of the z xomponent and the spherical component, where a,b and c are spherical components...look them up in a reference book.