Calculating Electric Field of a Solid Sphere with Nonuniform Charge Density

In summary, the conversation discusses a solid insulating sphere with nonuniform charge density and the calculation of charge Q_r contained within a radius r. The formula for finding this charge is Q_r = \pi ar^4, where a is a constant. The second part of the conversation involves finding the electric field (E) at a specific point (r=0.6m) using the equation E= Qr/4\pi E_0 R^3. Plugging in the previous answer for Q, which is \pi ar^4, gives the final equation for E as \pi ar^5/4\pi E_0 R^3. However, the calculation provided in the conversation results in the incorrect answer of
  • #1
Punchlinegirl
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0
Consider a solid insulating sphere of radius b with nonuniform charge density [tex] \sigma = ar [/tex] where a is a constant. Find the charge Q_r contained within the radius r, when r < b. Note: The volume element dV for a spherical shell of radius r and thickness dr is equal to [tex] 4\pi r^2 [/tex].
I got this part, the answer was Q_r = [tex] \pi ar^4 [/tex].
The second part says if a = 2 x 10^-6 C/m^4 and b= 1 m, find E at r= 0.6 m. Answer in units of N/C.
I used the equation [tex] E= Qr/4\pi E_0 R^3 [/tex]
Plugging in the previous answer for Q gave me [tex] E= (\pi ar^4)r/ 4\pi E_0 R^3 [/tex]
so [tex] \pi ar^5/4\pi E_0 B^3 [/tex] since R=B in this problem.
then [tex] \pi (2 x 10^-6)(.6)^5/ 4\pi (8.85 x 10^-12) [/tex]
This gave me 4378 N/C... which is wrong.
Help please?
 
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  • #2
[tex] E= Qr/4\pi E_0 R^3 [/tex]

Did you mean Q_r?
 
  • #3
Yes, sorry I meant Q_r * r.
 

1. How do you calculate the electric field of a solid sphere with nonuniform charge density?

To calculate the electric field of a solid sphere with nonuniform charge density, you can use the formula E = (1/4πε0) * ∫V ρ(r) * (r - r')/|r - r'|3 dτ, where ρ(r) represents the charge density at a given distance r from the center of the sphere and ε0 is the permittivity of free space. This integral takes into account the contribution of each infinitesimal charge element within the sphere to the overall electric field.

2. How does the electric field change as you move away from the center of the sphere?

The electric field of a solid sphere with nonuniform charge density will generally decrease as you move away from the center. This is because the charge density typically decreases as you move further from the center, resulting in a smaller overall contribution to the electric field from each infinitesimal charge element.

3. Can the electric field of a solid sphere with nonuniform charge density ever be zero?

Yes, it is possible for the electric field to be zero at certain points outside of the sphere. This can occur if the charge density is symmetrically distributed within the sphere, resulting in a cancellation of electric field contributions from different regions of the sphere.

4. How does the shape of the charge distribution affect the electric field of a solid sphere?

The shape of the charge distribution can greatly affect the electric field of a solid sphere. For example, a nonuniform charge distribution that is more concentrated towards the center of the sphere will result in a stronger electric field near the center compared to a distribution that is more spread out. Additionally, asymmetrical charge distributions can lead to uneven electric fields around the sphere.

5. What is the significance of calculating the electric field of a solid sphere with nonuniform charge density?

Calculating the electric field of a solid sphere with nonuniform charge density is important for understanding the behavior of electric fields in real-world situations. This can be applied to various scenarios, such as understanding the electric field inside a charged object or predicting the behavior of charged particles in the presence of this field.

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