How to Calculate Charge Content of a Sphere with Spherical Symmetry?

In summary, the conversation discusses determining the total charge content of a sphere with a charge distribution of spherical symmetry. The attempted solution uses spherical coordinates to define dV and integrates over the entire sphere, but the final result is incorrect. The error is likely in defining dV as (r d \varphi )(r d \theta) dr instead of just r^2 sin \theta d \varphi d \theta dr.
  • #1
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Homework Statement


A charge distribution with spherical symmetry has the density [tex] \rho = \rho _0 r/R [/tex] for 0≤ r ≤ R. Determine the total charge content of the sphere.

Homework Equations



[tex] \rho = Q / V [/tex]

The Attempt at a Solution



I started by thinking of the charge dQ of a small volume dV, since

[tex] \rho = dQ / dV [/tex]

I used spherical coordinates to define dV, and said that dV would be

[tex] dV = (r d \varphi )(r d \theta) dr[/tex]

Where [tex] \varphi [/tex] goes from 0 to 2*pi, [tex] \theta[/tex] goes from -pi/2 to pi/2, and r goes from 0 to R, thus covering the entire sphere.

So:

[tex] dQ = \rho dV = \rho r^2 d \varphi d \theta dr = \frac{\rho _0 r}{R} r^2 d \varphi d \theta dr = \frac{\rho _0 r^3}{R} d \varphi d \theta dr [/tex]

I then integrated over this as:

Q = [tex]\int ^ {2 \pi} _ {0} \int ^ {\pi /2} _ {-\pi /2} \int ^ {R} _ {0} \frac{\rho _0 r^3}{R} d \varphi d \theta dr [/tex]

[tex] Q = 2 \pi (\pi /2 + \pi /2) \frac{\rho _0 }{R} \int ^ {R} _ {0} r^3 dr = 2 \pi ^2 \frac{\rho _0 }{R} \frac{R^4}{4} = \frac{1}{2} \pi ^2 \rho _0 R^3[/tex]

And this is not correct, and I can't figure out where I've gone wrong. (It's supposed to be only [tex] \pi \rho _0 R^3[/tex]
 
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  • #2
consider any spherical shell inside the sphere of radius x(<R) and thicknedd dx
find charge on it and then integrate it dx from 0 to R
 
  • #3
Hannisch said:
I used spherical coordinates to define dV, and said that dV would be

[tex] dV = (r d \varphi )(r d \theta) dr[/tex]

Where [tex] \varphi [/tex] goes from 0 to 2*pi, [tex] \theta[/tex] goes from -pi/2 to pi/2, and r goes from 0 to R, thus covering the entire sphere.

Check dV. It is wrong.

ehild
 

1. What is charge distribution?

Charge distribution refers to the way in which electric charge is distributed or spread out over an object or within a material.

2. How is charge distribution measured?

Charge distribution can be measured using various techniques, such as Coulomb's law, electric field mapping, and capacitance measurements.

3. What factors affect charge distribution?

The factors that affect charge distribution include the amount of charge present, the distance between charges, and the material properties of the object or material.

4. What are the different types of charge distribution?

The two main types of charge distribution are discrete and continuous. Discrete charge distribution occurs when charges are located at specific points, while continuous charge distribution occurs when charges are spread out over a larger area.

5. How does charge distribution impact electric fields?

Charge distribution plays a crucial role in determining the strength and direction of electric fields. The electric field is stronger in regions with higher charge density and weaker in regions with lower charge density.

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