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**1. Homework Statement**

A conductor sphere of radius R without charge is floating half-submerged in a liquid with dielectric constant ##\epsilon_{liquid}=\epsilon## and density ##\rho_l##. The upper air can be considered to have a dielectric constant ##\epsilon_{air}=1##. Now an infinitesimal charge ##\delta Q## is added to the sphere. Find how much it will submerge or raise to the lowest order in ##\delta Q##.

**2. Homework Equations**

Poisson equation

$$\nabla^2 \Phi=\rho / \epsilon_0$$

Boundary conditions

$$\vec{n}\cdot (\vec{D_1}-\vec{D_2})=\rho_s$$

$$\vec{n}\times (\vec{E_1}-\vec{E_2})=0$$

Energy of electromagnetic fields

$$U=\frac{1}{2} \int \vec{D} \cdot \vec{E}$$

**3. The Attempt at a Solution**

Using Archimedes principle we can easily find the density of the sphere ##\rho_s## before adding the charge:

$$\rho_s (\frac{4}{3}\pi R^3)-\alpha_{before}(\rho_l (\frac{4}{3}\pi R^3))=0$$

Where ##\alpha_{before}=1/2## is the percentage sumerged before adding the charge. Thus:

$$\rho_s=\rho_l$$

After adding ##\delta Q##, it induces a field on the sphere. On Jackson's, there's an exercise which suggests that the electric field in the half-sumerged sphere is radial, hence we can posit:

$$E=E(r)$$

Since the charge is small, we can neglect it and use Laplace's equation for the potential:

$$\nabla^2 \Phi=0$$

And apply the boundary conditions assuming the constitutive equation for an homogeneous, linear, and isotropic liquid:

$$\vec{D}=\epsilon \vec{E}$$

From there I could find the electric and displacement fields ##E## and ##D## in the sphere to find the energy:

$$U(\delta Q)=\frac{1}{2} \int \vec{D} \cdot \vec{E}$$

Now since the energy is related to the force as,

$$\delta F=-\frac{\partial U (\delta Q)}{\partial dz}$$

I could find the force, reintroduce it in the equation of mechanical equilibrium as:

$$\rho_s (\frac{4}{3}\pi R^3)-\alpha_{after}(\rho_l (\frac{4}{3}\pi R^3))+\delta F=0$$

Where now ##\alpha_{after}## is the new percentage that the sphere is sumerged after adding ##\delta Q##. However, I'm stuck on whether this is a good idea or I'm just putting equations that aren't related. Moreover, if I apply the boundary conditions, how do I deal with the part where the sphere is half-sumerged in the liquid? Do I apply a boundary condition on the liquid surface too?

I actually found a similar problem here but it seems it was never properly solved: https://www.physicsforums.com/threads/conducting-sphere-floating-in-dielectric.303125/