Change in Electric Energy of Spheres

[DougD720]

Homework Statement

A heavy nuclei can be approximately considered as a spherical ball with uniform volume density  = 4/3 * 10^ 25 C/m3. If a nuclei with total charge Q = 92e splits into 2 equal spherical piece and they fly away. What will be the total electric energy change during this fission?

What I think it's asking for is the difference in energy between the 1 sphere and 2 sphere states.

Homework Equations

$W=(1/2)\int \rho V \text{d\tau }$

The Attempt at a Solution

My approach here (if I'm interpreting the problem correctly) would be to calculate the energy for one sphere with charge 92e and then calculate the energy for one sphere of a different radius with charge 92e (and then times by 2).

I'm having difficulty approaching how to calculate the energy for a sphere of charge:

dU = dq * V(dr)

$U = \int dq V(dr)$
$U = (1/4 \pi ε)\int dq * (dr/r)$

I'm having trouble with the limits of integration, I tried from Infinity to 0, split it into Infinity to R and then R to 0 but I got infinities in the answer. If someone could just point me along the right direction here for how to attack this problem I'd appreciate it, it seems easy I'm just doing something wrong or missing something.

Thanks!

 

[BruceW]

My approach here (if I'm interpreting the problem correctly) would be to calculate the energy for one sphere with charge 92e and then calculate the energy for one sphere of a different radius with charge 92e (and then times by 2).

I agree to first calculate the energy for one sphere with charge 92e. This will give you the energy for the '1 sphere state' as you call it. But then to calculate the energy of the '2 sphere state', you need to calculate the energy of a sphere with different radius and half of the original charge (since total charge must be conserved), and then times by 2 since there are now 2 spheres. See what I mean? I think maybe you just made a mistake in typing 92e again, but you meant to type 46e.

I'm having difficulty approaching how to calculate the energy for a sphere of charge:

dU = dq * V(dr)

$U = \int dq V(dr)$
$U = (1/4 \pi ε)\int dq * (dr/r)$

I'm not sure about the equation you are using here. in particular, what does V(dr) mean? Do you mean V(r) (i.e. V as a function of r)? Also, This looks like the equation for the energy required to move a point charge through an electric potential. But the equation you need to use is simply the one you have put in your 'relevant equations' section.

Keep in mind that using the equation in your 'relevant equations' section, the charge density is the final charge density of the object, so you would only need to do the integration up to the radius of the sphere, not outside the sphere. For a derivation of this equation, you can look up gravitational binding energy, since that is completely equivalent (but it is for gravity instead, of course).

So looking at the equation in your 'relevant equations' section, you will need to work out the electric potential inside a uniformly charged sphere. That's the first part of the question really, so have a go at that.

 

[rude man]

Homework Statement

A heavy nuclei can be approximately considered as a spherical ball with uniform volume density  = 4/3 * 10^ 25 C/m3. If a nuclei with total charge Q = 92e splits into 2 equal spherical piece and they fly away. What will be the total electric energy change during this fission?

What I think it's asking for is the difference in energy between the 1 sphere and 2 sphere states.

Homework Equations

$W=(1/2)\int \rho V \text{d\tau }$

The Attempt at a Solution

My approach here (if I'm interpreting the problem correctly) would be to calculate the energy for one sphere with charge 92e and then calculate the energy for one sphere of a different radius with charge 92e (and then times by 2).

I'm having difficulty approaching how to calculate the energy for a sphere of charge:

dU = dq * V(dr)

$U = \int dq V(dr)$
$U = (1/4 \pi ε)\int dq * (dr/r)$

Can't figure your integral I'm afraid.

Think of building up the charged sphere layer by thin spherical shell layer. So at any stage of this buildup you have a sphere of charge (4/3)πr³ρ and you're moving the next layer of thickness dr from ∞ to r where r is the radius from the sphere's center to the outermost layer built up to that point.

Then use the familiar expression W = kq₁q₂/r and you get your answer.

You could alternatively use the expression for E field energy by computing that expression for the charged sphere. Come to think of it, that's even simpler.

 

[SammyS]

Homework Statement

A heavy nuclei can be approximately considered as a spherical ball with uniform volume density  = 4/3 * 10^ 25 C/m3. If a nuclei with total charge Q = 92e splits into 2 equal spherical piece and they fly away. What will be the total electric energy change during this fission?

What I think it's asking for is the difference in energy between the 1 sphere and 2 sphere states.

Homework Equations

$W=(1/2)\int \rho V \text{d\tau }$

The Attempt at a Solution

My approach here (if I'm interpreting the problem correctly) would be to calculate the energy for one sphere with charge 92e and then calculate the energy for one sphere of a different radius with charge 92e (and then times by 2).

I'm having difficulty approaching how to calculate the energy for a sphere of charge:

dU = dq * V(dr)

$U = \int dq V(dr)$
$U = (1/4 \pi ε)\int dq * (dr/r)$

I'm having trouble with the limits of integration, I tried from Infinity to 0, split it into Infinity to R and then R to 0 but I got infinities in the answer. If someone could just point me along the right direction here for how to attack this problem I'd appreciate it, it seems easy I'm just doing something wrong or missing something.

Thanks!

The plural of nucleus is nuclei. The singular of nuclei is nucleus.

For limits of integration:

How large is each nucleus?

You're given the charge density and the total charge. Putting those together, figure out the radius of each nucleus.

Find the self-energy of the Q = 92e nucleus.

Find the self-energy of each of the Q = 46e nuclei. Add these together.

What's the difference in initial self-energy and the final (include both nuclei) self-energy?

Where did that energy go?

 

[paranormal]

It seems like you are on the right track with your approach. To calculate the energy for a sphere of charge, you can use the equation W=(1/2)QV, where Q is the charge and V is the potential. Since the charge is given as 92e, you can simply plug that into the equation. The potential can be calculated using the formula V=kQ/r, where k is the Coulomb constant and r is the radius of the sphere. Since the problem states that the spheres have equal charge and are split into two equal pieces, you can use this formula for both spheres and then multiply the total energy by 2. The limits of integration for the potential would be from the radius of the sphere to infinity, since the potential at infinity is considered to be zero. I hope this helps guide you in the right direction.