What is the total charge of a solid sphere with given charge density?

In summary, the conversation discusses finding the total charge contained by a solid sphere with a given charge density (\rho=14.1\frac{pC}{m^{3}}\frac{r}{R}) and radius (R=5,6cm). The formula \rho=\frac{dq}{dV} is used to calculate the charge, with dq=\rho dV being the initial attempt. The conversation then moves on to discussing the use of volume integration in spherical coordinates to find the charge, but ultimately determines that the correct formula is q=14.1\frac{pC}{m^{3}} \pi r^{3}.
  • #1
Uku
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Homework Statement


I am given a charge density for a solid sphere
[tex]\rho=14.1\frac{pC}{m^{3}}\frac{r}{R}[/tex]
The r is the distance from the center of the sphere and R is the radius of the whole thing.

[tex]R=5,6cm[/tex]

Now I am asked for the whole charge contained by the sphere.

Homework Equations



[tex]\rho=\frac{dq}{dV}[/tex]

The Attempt at a Solution


[tex]dq=\rho dV[/tex]
[tex]dq=4.1\frac{pC}{m^{3}}\frac{r}{R} dV[/tex]
I'll just denote the picocoulomb into B
[tex]q=\frac{B}{R} \int r dV[/tex]

Right, here I land. This is from Halliday, second year thing, I bet they don't expect you to do volume integration in spherical coordinates or anything such. I could write it:
[tex]dV=\frac{4}{3} \pi dr^{3}[/tex]?

Pff...

EDIT:

Ok, now I get it I think:
[tex]q=\frac{B}{R} \int r dV[/tex]
is actually
[tex]q=B \int dV[/tex]
[tex]q=14.1\frac{pC}{m^{3}} \frac{4}{3} \pi r^{3}[/tex]

ought to give me the right answer

EDIT:

It does not.
The right answer is given by
[tex]q=14.1\frac{pC}{m^{3}} \pi r^{3}[/tex]

But how do I land that?
 
Last edited:
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  • #2
just consider a small spherical shell of radius x(<R) and thickness dx

write dq for it (by finding volume of "shell" --- not of sphere of radius x----) and integrate it from x=0 to x=R
 

FAQ: What is the total charge of a solid sphere with given charge density?

1. What is the formula for calculating the total charge of a sphere?

The total charge of a sphere can be calculated using the formula Q = 4πε0r2E, where Q is the total charge, ε0 is the permittivity of free space, r is the radius of the sphere, and E is the electric field strength at the surface of the sphere.

2. How does the total charge of a sphere affect its electric field?

The total charge of a sphere directly affects its electric field. The electric field strength at the surface of the sphere is directly proportional to the total charge and inversely proportional to the radius of the sphere. This means that as the total charge increases, the electric field strength also increases.

3. Can the total charge of a sphere be negative?

Yes, the total charge of a sphere can be negative. This would indicate that the sphere has an excess of electrons, making it negatively charged. The electric field of a negatively charged sphere would point towards the center of the sphere.

4. How does the total charge of a sphere affect its potential energy?

The total charge of a sphere affects its potential energy through the electric potential energy equation, U = QV, where U is the potential energy, Q is the total charge, and V is the electric potential. As the total charge increases, the potential energy also increases, and vice versa.

5. Can the total charge of a sphere be changed?

Yes, the total charge of a sphere can be changed. This can be done by adding or removing electrons from the sphere, which will result in a change in the total charge. The electric field and potential energy of the sphere will also be affected by this change in charge.

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