- #1
Cedric Chia
- 22
- 2
Homework statement:
Find the electric field a distance z from the center of a spherical shell of radius R that carries a uniform charge density σ.
Relevant Equations: Gauss' Law
$$\vec{E}=k\int\frac{\sigma}{r^2}\hat{r}da$$
My Attempt:
By using the spherical symmetry, it is fairly obvious that the horizontal component of electric field at z cancel, leaving only $$\vec{E}=E_{z}\hat{z}$$
and from there we can get rid of the complexity of the $$\hat{r}$$
and continue... BUT !
What if I'm not aware of this symmetry property of the spherical shell and must do the parameterization? I know the x component and y component of this vector: $$\vec{r}=<-Rsin\phi cos\theta ,-Rsin\phi sin\theta, zcos\phi>$$
is correctly parameterized in this way but the z component doesn't seem right...
i.e. when:$$\theta=0$$ $$\phi=180$$
the z component should be:
$$R+\text{distance from origin to the point outside the sphere}$$
in order to get from the bottom of the sphere to the top and some extra distance to get to the point
What is the z component in this parameterization? Please HELP
Find the electric field a distance z from the center of a spherical shell of radius R that carries a uniform charge density σ.
Relevant Equations: Gauss' Law
$$\vec{E}=k\int\frac{\sigma}{r^2}\hat{r}da$$
My Attempt:
By using the spherical symmetry, it is fairly obvious that the horizontal component of electric field at z cancel, leaving only $$\vec{E}=E_{z}\hat{z}$$
and from there we can get rid of the complexity of the $$\hat{r}$$
and continue... BUT !
What if I'm not aware of this symmetry property of the spherical shell and must do the parameterization? I know the x component and y component of this vector: $$\vec{r}=<-Rsin\phi cos\theta ,-Rsin\phi sin\theta, zcos\phi>$$
is correctly parameterized in this way but the z component doesn't seem right...
i.e. when:$$\theta=0$$ $$\phi=180$$
the z component should be:
$$R+\text{distance from origin to the point outside the sphere}$$
in order to get from the bottom of the sphere to the top and some extra distance to get to the point
What is the z component in this parameterization? Please HELP