electric flux = [MATH]\Phi_E = 4\pi r^2E[/MATH]
where [MATH]E[/MATH] is the electric field and [MATH]4\pi r^2[/MATH] is the surface area of the sphere
When I solve the problem with surface integral, I don’t get the same results. Why? What did I do wrong?
This is what I did.
According to Gauss Law, the electric flux is
[MATH]\Phi_E = \int \vec{E} \cdot \vec{dA}[/MATH]
[MATH]\vec{E}[/MATH] points radially out of the sphere and it is constant
the integral will become
[MATH]\Phi_E = E\int \vec{dA}[/MATH] = [MATH]\int \int \frac{\vec{n}}{|\vec{n}|} \ |\vec{n}| \ dx \ dy = E\int \int \vec{n} \ dx \ dy[/MATH]
where [MATH]\vec{n}[/MATH] is the normal vector
sphere equation (taking the upper half) is
[MATH]f(x,y) = \sqrt{r^2 - x^2 - y^2}[/MATH]
[MATH]\vec{n} = <-f_x, -f_y, 1>[/MATH]
[MATH]\Phi_E = 2E\int \int \frac{x}{\sqrt{r^2 - x^2 - y^2}} + \frac{y}{\sqrt{r^2 - x^2 - y^2}} + 1 \ dx \ dy[/MATH]
Solving this integral gives
[MATH]\Phi_E = 2\pi r^2 E[/MATH]
Why?
where [MATH]E[/MATH] is the electric field and [MATH]4\pi r^2[/MATH] is the surface area of the sphere
When I solve the problem with surface integral, I don’t get the same results. Why? What did I do wrong?
This is what I did.
According to Gauss Law, the electric flux is
[MATH]\Phi_E = \int \vec{E} \cdot \vec{dA}[/MATH]
[MATH]\vec{E}[/MATH] points radially out of the sphere and it is constant
the integral will become
[MATH]\Phi_E = E\int \vec{dA}[/MATH] = [MATH]\int \int \frac{\vec{n}}{|\vec{n}|} \ |\vec{n}| \ dx \ dy = E\int \int \vec{n} \ dx \ dy[/MATH]
where [MATH]\vec{n}[/MATH] is the normal vector
sphere equation (taking the upper half) is
[MATH]f(x,y) = \sqrt{r^2 - x^2 - y^2}[/MATH]
[MATH]\vec{n} = <-f_x, -f_y, 1>[/MATH]
[MATH]\Phi_E = 2E\int \int \frac{x}{\sqrt{r^2 - x^2 - y^2}} + \frac{y}{\sqrt{r^2 - x^2 - y^2}} + 1 \ dx \ dy[/MATH]
Solving this integral gives
[MATH]\Phi_E = 2\pi r^2 E[/MATH]
Why?