How to Find the Total Charge from its Density?

[JasonHathaway]

Homework Statement

A ball whose its center on the origin, with radius of 0.2m, and contains a charge density $\frac{2}{\sqrt{x^{2}+y^{2}}}c/m^{3}$

Homework Equations

$D=\frac{\psi}{S}=\frac{\psi}{4\pi r^{2}}$

Where $\psi=Q$.

The Attempt at a Solution

The density is $c/m^{3}$, which is volume not surface.
Anyway, I've used the previous equation $\frac{2}{\sqrt{x^{2}+y^{2}}}=\frac{Q}{4×\pi ×0.2^{2}} \rightarrow Q=\frac{2×4×\pi×0.2^{2}}{\sqrt{x^{2}+y^{2}}}$

Is that right?

 

[Simon Bridge]

You have to integrate the density over the sphere volume.

 

[JasonHathaway]

You mean $\int\limits_0^{2\pi} \int\limits_0^ \pi \int\limits_0^{0.2} \frac{2}{\sqrt{x^{2}+y^{2}}} r^{2} sin\theta dr d\theta d\phi$?

 

[CAF123]

The density is $c/m^{3}$, which is volume not surface.
Anyway, I've used the previous equation $\frac{2}{\sqrt{x^{2}+y^{2}}}=\frac{Q}{4×\pi ×0.2^{2}} \rightarrow Q=\frac{2×4×\pi×0.2^{2}}{\sqrt{x^{2}+y^{2}}}$

That formula would give you a charge per unit area and is only applicable when the density is uniform.

You mean $\int\limits_0^{2\pi} \int\limits_0^ \pi \int\limits_0^{0.2} \frac{2}{\sqrt{x^{2}+y^{2}}} r^{2} sin\theta dr d\theta d\phi$?

Yes, $Q = \int \rho\,\text{d}V$ in general and only when $\rho$ is not dependent on V (i.e is uniform throughout the volume) does this simplify to $Q = \rho \int \, \text{d}V = \rho V$. But you should express your integrand in spherical polar coordinates.

 

[ehild]

You mean $\int\limits_0^{2\pi} \int\limits_0^ \pi \int\limits_0^{0.2} \frac{2}{\sqrt{x^{2}+y^{2}}} r^{2} sin\theta dr d\theta d\phi$?

yes, but express x²+y² also in terms of spherical coordinates.

ehild

 

[Simon Bridge]

That's what I had in mind - yep.
The symmetry of the density function suggests cylindrical-polar may be better.
Maybe look at the shell method?

in general, the volume element $\text{d}V$ at position $\vec r$ has charge $\text{d}q=\rho(\vec r)\text{d}V$

The total charge $Q$ in a volume $V$ is the sum of all those $\text{d}q$ elements: $Q=\int_V\text{d}q$

Always start with a sketch - try to understand the density function: where is it a maximum, where a minimum, how does it change, etc?