Determining Charge from Charge Density

In summary, the problem involves a positively charged nonconducting solid sphere with a nonuniform volume charge density. To determine the charge on the sphere, the integral of the charge density must be taken over the volume of the sphere. The correct solution involves two integrals, one from 0 to R/2 and one from R/2 to R, with the corresponding charge densities. The attempt at the solution appears to be missing a factor in the second integral.
  • #1
Sunbodi
22
0

Homework Statement


A positively charged nonconducting solid sphere of radius R has a nonuniform volume charge density given by ρ0 for r≤R/2 and given by 2ρ0(1−r/R) for R/2 ≤r≤R, where r is the radial distance from the sphere center.Part A
Determine the charge q on the sphere in terms of ρ0 and R.

Homework Equations



∫ρ*dv = Q0R/2 ρ0 dVolume + R/2R ρ0 dVolume = Q from 0 to R/2 + Q from R/2 to R

The Attempt at a Solution


0R/2 ρ0 dVolume = 0R/2 ρ0 4πr2 dr
= R3πρ0/2R/2R ρ0 dVolume = R/2R ρ0 4πr2 dr

= (4R3- 3R4)*8πρ0/(12R)

Is my math wrong or is my attempt at the solution wrong? If both is wrong could you help me more with the attempt at the solution?

Thanks!

Sun

(ρ) Is rho btw.
 
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  • #2
What is ∫r2.dr?

In the second integral, you left out the (1-r/R) factor.
 

What is charge density?

Charge density refers to the amount of electric charge per unit volume of a material. It is typically denoted by the symbol ρ and is measured in coulombs per cubic meter (C/m³).

How is charge density related to electric charge?

Charge density and electric charge are directly proportional to each other. This means that as the amount of electric charge increases, the charge density also increases.

How do you determine charge from charge density?

To determine the total electric charge from charge density, you need to multiply the charge density (ρ) by the volume (V) of the material. This can be expressed as Q = ρV. The resulting charge will have the unit of coulombs (C).

What is the significance of determining charge from charge density?

Determining charge from charge density is important in understanding and analyzing electric fields and their effects. It allows scientists to accurately measure the amount of electric charge present in a material or system.

Are there any limitations to determining charge from charge density?

Yes, there are limitations to determining charge from charge density. This method assumes that the material has a uniform charge distribution, which may not always be the case. Additionally, the accuracy of the measurement can be affected by external factors such as electric fields from other sources.

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