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## Homework Statement:

- Find the electric field a distance z from the center of a spherical surface of radius R which carries a uniform charge density ##\sigma##. Hint: use the law of cosines to write r in terms of R and ##\theta## where ##\theta## is inclination/elevation angle of sphere.

## Relevant Equations:

- Coulomb's Law: $$dE=\frac{1}{4 \pi \epsilon_0} \frac{dq}{r^2}$$

Well, I really don't understand what is the use of the hint.

I try to solve this problem with Coulomb's Law and try to do in spherical coordinates and got very messy infinitesimal field due to the charge of infinitesimal surface element of the sphere.

Here what I got:

$$\vec{r}=\vec{r_P} + \vec{R}$$ where ##\vec{r}## is position of the point of field (P in question) relative to infinitesimal surface element; ##\vec{r_P}## is position of the point of field relative to the origin, and ##\vec{R}## is position of vector radius of the sphere; as shown in figure.

I get, $$\vec{r} = (z cos \theta - R)\hat{r} - z sin\theta \hat{\theta}$$.

With the domain of integration ##0<=\theta<=\pi## ##0<=\\phi<=2\pi##; how I can solve the integration and find the electric field?

Second, is there another method which is easier? Thankss

I try to solve this problem with Coulomb's Law and try to do in spherical coordinates and got very messy infinitesimal field due to the charge of infinitesimal surface element of the sphere.

Here what I got:

$$\vec{r}=\vec{r_P} + \vec{R}$$ where ##\vec{r}## is position of the point of field (P in question) relative to infinitesimal surface element; ##\vec{r_P}## is position of the point of field relative to the origin, and ##\vec{R}## is position of vector radius of the sphere; as shown in figure.

I get, $$\vec{r} = (z cos \theta - R)\hat{r} - z sin\theta \hat{\theta}$$.

With the domain of integration ##0<=\theta<=\pi## ##0<=\\phi<=2\pi##; how I can solve the integration and find the electric field?

Second, is there another method which is easier? Thankss