Electrostatic Energy of a solid sphere with a cavity

In summary, the conversation discusses using the formula ##W = ε_0/2 \int E^2d\tau## for all space to find the electric field ##E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}##, where ##\rho## is the charge density. The expression for E is applicable for ##r>R## but also needs to be found for ##b<r<R##. The total potential energy can be calculated using the formula $$U = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{R}^{\inf}1/r^2dr +
  • #1
Arman777
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Homework Statement
Determine the electrostatic energy U stored in the sphere
Relevant Equations
##W = ε_0/2 \int E^2d\tau## for all space
1571252938043.png

I tried to use ##W = ε_0/2 \int E^2d\tau## for all space. So I find that ##E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}## where ##\rho## is the charge denisty. So from here when I plug the equation I get something like

$$W = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{?}^{\inf}1/r^2dr$$

Is this approach correct ?

At this point I get stuck for the boundry conditions. If I put R I get something meaningless
 
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  • #2
Arman777 said:
I tried to use ##W = ε_0/2 \int E^2d\tau## for all space.
OK
So I find that ##E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}## where ##\rho## is the charge denisty.
For what region of space is this expression for E applicable? You have three distinct regions to consider.
 
  • #3
TSny said:
OK
For what region of space is this expression for E applicable? You have three distinct regions to consider.
This is for ##r>R## so I have to also find ##E(r<b)##, ##E(b<r<R)## ?

Then The U will be,

$$U = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{R}^{\inf}1/r^2dr + \frac{e_0}{2} \int_{b}^{R}\frac{\rho (r - b^3)}{3ε_0}dr$$

Where ##E = \frac{\rho (r - b^3)}{3ε_0}, (b<r<R)##
 
  • #4
I think the correct expression for the E-field for ##b<r<R## is $$E=\frac{\rho(r^3-b^3)}{3\epsilon_0r^2}$$.
 

Related to Electrostatic Energy of a solid sphere with a cavity

1. What is electrostatic energy?

Electrostatic energy is the potential energy stored in a system of charged particles due to their interactions with each other.

2. How is electrostatic energy calculated?

The electrostatic energy of a solid sphere with a cavity can be calculated using the formula E = (3/5) * (k * Q^2)/(R + r), where E is the electrostatic energy, k is the Coulomb constant, Q is the charge of the sphere, R is the radius of the sphere, and r is the radius of the cavity.

3. What factors affect the electrostatic energy of a solid sphere with a cavity?

The electrostatic energy of a solid sphere with a cavity is affected by the charge of the sphere, the size of the sphere and the cavity, and the distance between the charged particles within the system.

4. How does the presence of a cavity affect the electrostatic energy of a solid sphere?

The presence of a cavity in a solid sphere can significantly decrease the electrostatic energy as it creates a void that reduces the interactions between charged particles, resulting in a lower overall energy.

5. What is the significance of studying the electrostatic energy of a solid sphere with a cavity?

Understanding the electrostatic energy of a solid sphere with a cavity is important in various fields such as materials science, physics, and engineering. It allows for the prediction and manipulation of the electrostatic properties of materials, which is crucial in the development of new technologies and applications.

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