Charge Distribution from Known Potential - Induced on Surface of Metal Sphere

[bjnartowt]

Homework Statement

Using the method of images: discuss the problem of a point-charge “q” inside a hollow, grounded, conducting-sphere-shell of inner radius “a”. That charge "q" is located at $${\left| {{{{\bf{\vec r}}}_0}} \right|}$$. We don't care about its angular position. Oh yes: the problem is two dimensional: as you will see, potential is termed as r, theta dependent.

(b) Find the induced surface-charge density. The potential is known to be:
$$\Phi (r,\theta ) = \frac{q}{{4\pi {\varepsilon _0}}}\left( {\frac{1}{{\sqrt {{r^2} + {{\left| {{{{\bf{\vec r}}}_0}} \right|}^2} - 2r\left| {{{{\bf{\vec r}}}_0}} \right|\cos \theta } }} - \frac{a}{{\sqrt {{r^2}{{\left| {{{{\bf{\vec r}}}_0}} \right|}^2} + {a^4} - 2r\left| {{{{\bf{\vec r}}}_0}} \right|{a^2}\cos \theta } }}} \right)$$

Homework Equations

The Attempt at a Solution

Surface charge density: it’s derived from integral Gauss’s Law:
$$\int {{\bf{\vec E}} \bullet d{\bf{\vec A}}} = \frac{Q}{{{\varepsilon _0}}}{\rm{ }} \to {\rm{ }}{\bf{\vec E}}(r,\theta ) = \frac{Q}{{{\varepsilon _0}\int {d{\bf{\vec A}}} }} = \frac{{\sigma (r,\theta )}}{{{\varepsilon _0}}}$$ [I.2]

And: curl-less-ness of electric field:
$${\bf{\vec E}} = - \vec \nabla \Phi$$ [I.3]

Then: [I.2] and [I.3] give the surface-charge density as a function of the potential we derived as:
$$\sigma = - {\varepsilon _0}\left| {\vec \nabla \Phi } \right|$$

The spherical gradient operator reduced to polar coordinates:
$$\vec \nabla = {\bf{\hat r}}\frac{\partial }{{\partial r}} + \hat \theta \frac{1}{r}\frac{\partial }{{\partial \theta }}$$

Umm…I feel like I’m doing this really wrong. Never have I needed to use the "theta-hat" in computation…only in conceptually grasping the field of a dipole.

Then: someone tells me to use:
$$\sigma = {\left. {{\varepsilon _0}\frac{{\partial \Phi (r,\theta )}}{{\partial r}}} \right|_{r = a}}$$

But I disagree: neither the charge distribution nor the potential are symmetric in "theta". Surely: as you "tilt through theta" away from the charge hovering outside the metal sphere: the charge will go from positive to negative when you've gone from theta = 0 (right under the charge) to theta = pi (on the opposite pole of the metal-sphere, away from the charge)?

 

[gabbagabbahey]

Surface charge density: it’s derived from integral Gauss’s Law:
$$\int {{\bf{\vec E}} \bullet d{\bf{\vec A}}} = \frac{Q}{{{\varepsilon _0}}}{\rm{ }} \to {\rm{ }}{\bf{\vec E}}(r,\theta ) = \frac{Q}{{{\varepsilon _0}\int {d{\bf{\vec A}}} }} = \frac{{\sigma (r,\theta )}}{{{\varepsilon _0}}}$$ [I.2]

No, $\int\textbf{E}\cdot d\textbf{A} = \textbf{E}\cdot\int d\textbf{A}$ not $\textbf{E}\int d\textbf{A}$, and only in the case where $\textbf{E}$ is uniform/constant over the surface. And, $\textbf{a}\cdot\textbf{b}= c$ does not mean that $$\textbf{a}=\frac{c}{\textbf{b}}$$

Then: someone tells me to use:
$$\sigma = {\left. {{\varepsilon _0}\frac{{\partial \Phi (r,\theta )}}{{\partial r}}} \right|_{r = a}}$$

But I disagree: neither the charge distribution nor the potential are symmetric in "theta". Surely: as you "tilt through theta" away from the charge hovering outside the metal sphere: the charge will go from positive to negative when you've gone from theta = 0 (right under the charge) to theta = pi (on the opposite pole of the metal-sphere, away from the charge)?

"someone" is correct. The equation comes from the boundary conditions on the $\textbf{E}$-field as it crosses a surface. The component of the normal $\textbf{E}$-field that is perpendicular to the surface will be discontinuous by an amount $\frac{\sigma}{\epsilon_{0}}$ (this is likely derived in your textbook directly from Gauss' Law, and I strongly recommend you read that section of your text).

 

[bjnartowt]

I think I see it. The field discontinuity across a charged surface is:
${{\bf{\vec E}}_{above}} - {{\bf{\vec E}}_{below}} = \sigma /{\varepsilon _0}$

...in which you use the field/potential relation:
${\bf{\vec E}} = - {\mathop{\rm grad}\nolimits} (\Phi )$

...to get what "someone" told me:
$\sigma = - {\varepsilon _0}\frac{{\partial \Phi }}{{\partial r}}$

...and I'm just the short distance of "plug-n'-chug" away from a possibly-correct charge density. No?