Force between point charges at the center of two spherical shells

[vcsharp2003]

Homework Statement

Let charges of $q_1$ and $q_2$ be placed at the center of two spherical shells of radii $r_1$ and $r_2$ respectively. If these spherical shells are placed so that their centers are a distance $d$ $(d > (r_1 + r_2))$ apart, then what is the force between the point charges $q_1$ and $q_2$?

Relevant Equations

$F =\dfrac { k q_1 q_2} {r^2}$, which is Coulomb's Law in Electrostatics (k is a constant of proportionality)

If these point charges were placed in vacuum without any spherical shells in the picture, then the force between these charges would be $F =\dfrac { k q_1 q_2} {d^2}$.

But, I am unable to reason how spherical shells would alter the force between them.

I do know that if charges were on the spherical shells rather than at the centers, then we would consider the charge on each spherical shell to be at its center when calculating the electrostatic force between the spherical shells. But, in this problem the spherical shells carry no charge.

 

[haruspex]

Homework Statement:: Let charges of $q_1$ and $q_2$ be placed at the center of two spherical shells of radii $r_1$ and $r_2$ respectively. If these spherical shells are placed so that their centers are a distance $d$ $(d > (r_1 + r_2))$ apart, then what is the force between the point charges $q_1$ and $q_2$?
Relevant Equations:: $F =\dfrac { k q_1 q_2} {r^2}$, which is Coulomb's Law in Electrostatics (k is a constant of proportionality)

I am unable to reason how spherical shells would alter the force between them.

Quite. There is no mention of the shells being conductors or dielectrics. Even if they were, there is no change to the forces the point charges exert on each other; the induced charges/polarisation can be thought of as adding forces rather than as altering the direct forces.

 

[vcsharp2003]

Quite. There is no mention of the shells being conductors or dielectrics. Even if they were, there is no change to the forces the point charges exert on each other; the induced charges/polarisation can be thought of as adding forces rather than as altering the direct forces.

If the spherical shells were conductors then my guess is that electric lines of force would be more spread out within the shell part as compared to the same lines of force in vacuum space. But why would the shells appearing between the point charges not affect the force between them?

Also, on another line of thinking, one could look at the space between the straight line joining the point charges as a composite medium space i.e. multiple materials are placed between the two point charges. If we can then find the equivalent dielectric constant then we can use it in Coulomb's law. The space between the point charges would be in the following physical order: vacuum, spherical shell, vacuum, spherical shell, vacuum. For vacuum we could use $K_1 = 1$. After we determine the equivalent dielectric constant $K$ we could use the formula $F = \dfrac {kq_1q_2}{Kr^2}$, though I am not sure if the above approach is valid in this problem.

 

[haruspex]

If the spherical shells were conductors then my guess is that electric lines of force would be more spread out

The net field would be different, but it is perfectly valid to think of that as the sum of the field due to the point charged (unchanged) and the field due to the induced charge (or, for the dielectric, induced polarisation). The question asks for the force between the particles, not the net force on the particles. As I see it, it is perfectly reasonable to argue that the force between them is unaffected.

 

[vcsharp2003]

The question asks for the force between the particles, not the net force on the particles.

Yes, that sounds valid. From Superposition principle, we know that two charges exert force on each other independent of the other charges in their neighborhood. Here we have the charge distribution due to induction on each conducting spherical shell exerting a force on the point charges and also there is electrostatic force between the point charges. All these forces would act independent of each other according to Superposition principle.