- #1
Hamal_Arietis
- 156
- 15
- Homework Statement
- Find the electric potential at a height d above the center of a square sheet
(side 2a) carrying a uniform surface charge σ.
- Relevant Equations
- [tex] σdS=σdxdy[/tex]
[tex] V=\int \frac{\sigma dS}{r}[/tex]
I found out the equation of electric potential, that is
[tex]V=\int_{-a}^a \int_{-a}^{a} \frac{σdxdy}{4 \pi \epsilon_0\sqrt{x^2+y^2+d^2}}=\int_{0}^a \int_{0}^{a} \frac{σdxdy}{\pi \epsilon_0\sqrt{x^2+y^2+d^2}}[/tex]
but I couldn't calculate the integral.
It seems convenient if we use the polar coordinate, that is, assume that
[tex]x=rcos\theta, y=rsin\theta[/tex]
then
[tex]\frac{\pi\epsilon_0 V}{\sigma}=\int_{0}^{\pi/4} \int_{0}^{\frac{a}{cos\theta}} \frac{rdrd\theta}{\sqrt{r^2+d^2}}+\int_{\pi/4}^{\pi/2} \int_{0}^{\frac{a}{sin\theta}} \frac{rdrd\theta}{\sqrt{r^2+d^2}}[/tex]
But I find it difficult to calculate.
I also think another method that is find the electric field $$\vec E$$ then
[tex] V=\int \vec E \vec dl[/tex]
but it seems hard to integrate too .
[tex]V=\int_{-a}^a \int_{-a}^{a} \frac{σdxdy}{4 \pi \epsilon_0\sqrt{x^2+y^2+d^2}}=\int_{0}^a \int_{0}^{a} \frac{σdxdy}{\pi \epsilon_0\sqrt{x^2+y^2+d^2}}[/tex]
but I couldn't calculate the integral.
It seems convenient if we use the polar coordinate, that is, assume that
[tex]x=rcos\theta, y=rsin\theta[/tex]
then
[tex]\frac{\pi\epsilon_0 V}{\sigma}=\int_{0}^{\pi/4} \int_{0}^{\frac{a}{cos\theta}} \frac{rdrd\theta}{\sqrt{r^2+d^2}}+\int_{\pi/4}^{\pi/2} \int_{0}^{\frac{a}{sin\theta}} \frac{rdrd\theta}{\sqrt{r^2+d^2}}[/tex]
But I find it difficult to calculate.
I also think another method that is find the electric field $$\vec E$$ then
[tex] V=\int \vec E \vec dl[/tex]
but it seems hard to integrate too .
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