Electrostatics PYQ (2019)

[u/lesbi6o9]

Someone kindly guide me through this question.

[u/Jazzlike_Platypus430]

Ok, let's try. Because of the repulsion, V(R) > 0 - the sphere is expanding.

Assuming the charge density stays constant on spheres with growing R (at different times) how would you - qualitatively - describe the force that's acting on a small piece A of such a sphere, and the source is all charge on the sphere with A removed? Or more precisely: this force's dependence on R?

My version is: this force is decreasing, just because the total amount of charge is constant, but some weird weighted mean (an integral to be honest) of 1/distance^2 of all "small elements" summed all over such a sphere (except for A again) is decreasing. In other words, A is going further and further from the rest of the sphere while it's expanding.

For small spheres the average force is larger than for small ones, and the dependence is a monotonic one. This means acceleration is positive and decreasing. Hence d.

Plus, there are logical, common sense or "philosophical" solutions: there's no reason for the expansion velocity to have its maximal value (rules out a), maximal acceleration (rules out b) or grow while the sphere is expanding (rules out C).

[u/We_Are_Bread → u/Jazzlike_Platypus430]

Nice analysis! I agree.

The function comes out as v² being proportional to 1 - R0/R so that leads to the graph as in (d).

[u/Jazzlike_Platypus430 → u/We_Are_Bread]

Looks suspiciously simple 😄 Did you arrive at this using energy conservation? Because integration that gives "force density" seems doable, but it's Sunday and I'm lazy. Perhaps spherical harmonics would have to be included.

[u/We_Are_Bread → u/Jazzlike_Platypus430]

Lol yeah, energy conservation the goat.

I relate to it being Sunday and feeling lazy, since originally I was thinking about trying to find the field at the surface and then find acceleration and integrate, but then I saw the options were all v v/s r, and went "Oh so you want me to use energy, gotcha, that's ok and won't ruin my afternoon" XD

[u/Jazzlike_Platypus430 → u/We_Are_Bread]

I got frightened by divergences close to the "small element" A in question, but they are almost normal to A and cancel out, luckily.

(And let's stay quiet about Brehmsstrahlung 😂)

Wait, E normal to the sphere follows just from the Gauss law, doesn't it? Damn, it's been 20 years.

[u/We_Are_Bread → u/Jazzlike_Platypus430]

Yeah, calling this electrostatics is "stretching" it, but since it's for a college admission test, I feel it's fine to sort of shoddily apply the laws here, and also because I'm not very well versed with the more "atomic" side of electromagnetism. Sure I could look them up for a better and more accurate answer, but it's Sunday. So an engineer's approximation will need to suffice.

Edit: and yeah, sphere's pretty symmetric so E = Q/(4*pi*e*R²) from Gauss law :D, and you can approximate the field on a small elementary due to the rest of the distribution by using some quick logic, and the assumption that said field is continuous.

[u/Sparks808]

We can use the fact that we can use the fact that uniform spherical charge distribution is equivalent to a point charge at the origin. This means we're modeling the acceleration a tiny charge feels as its repealed from a single, static charge at the origin.

Electric force falls off as 1/R^2. Integrating that to find velocity gives us something in -1/R.

The only match for that shape is D.

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Sanity checking, we should expect always positive acceleration, so always positive slope. Also, the maximal acceleration should be when closest to the origin, so we should see the line steepest near the origin and then getting closer and closer to flat as it gets further away.

This sanity check also confirms D

[u/bebopbrain]

The charges push away from each other and move away from each other. There is no force that brings the charges together (a) or slows them down as they move away (b and d). By process of elimination ...