Calculate the electrostatic energy of a homogeneously charged sphere

In summary, the electrostatic energy of a homogeneously charged sphere of volume V can be calculated by finding the work done in bringing a charge to the sphere with the known charge distribution. This can be compared with 2 times the electrostatic energy of a homogeneously charged sphere with half the volume. The limits for the integration would be from 0 to the radius of the sphere.
  • #1
Quelsita
49
0
QUESTION:

Calculate the electrostatic energy of a homogeneously charged sphere of Volume V and
compare the result with 2 times the electrostatic energy of a homogeneously charged
sphere of V/2.

SOLUTION:

OK, so we have a charge Q which is uniformly distributed within a sphere of radius, R.

We know:
q(r) is the charge in the sphere when it has
attained radius, r.
q(r)= [tex]\rho[/tex](4/3)[tex]\Pi[/tex]r3

the work done in bringing a charge dq to it is dW
dW= (q(r) dq)/(4[tex]\Pi[/tex]r[tex]\epsilon[/tex]0

dq= [tex]\rho[/tex]4[tex]\Pi[/tex]r2dr

so, we plug in the knowns:

dW= ([tex]\rho[/tex](4/3)[tex]\Pi[/tex]r3 * [tex]\rho[/tex]4[tex]\Pi[/tex]r2dr )/ (4[tex]\Pi[/tex]r[tex]\epsilon[/tex]0)

and integrate.

Is this correct thus far?
What are the limits?

Thanks!
 
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  • #2
Your bounds would be 0 to R as that covers the whole sphere.
 

What is electrostatic energy?

Electrostatic energy is the potential energy associated with the interaction between electric charges at rest.

What is a homogeneously charged sphere?

A homogeneously charged sphere is a spherical object in which the charge is distributed evenly throughout its volume.

How is the electrostatic energy of a homogeneously charged sphere calculated?

The electrostatic energy of a homogeneously charged sphere can be calculated using the formula U = (3/5)*(Q^2)/r, where U is the electrostatic energy, Q is the charge of the sphere, and r is the radius of the sphere.

Can the electrostatic energy of a homogeneously charged sphere be negative?

No, the electrostatic energy of a homogeneously charged sphere cannot be negative. It is always a positive value as like charges repel each other, leading to a positive potential energy.

What are the units of electrostatic energy?

The units of electrostatic energy are joules (J) in the SI system. In CGS units, the unit is ergs (erg).

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