Step-by-step Derivations

1. Setup and Coordinate System Let's place the center of the smaller sphere (Sphere 1) at the origin $(0,0,0)$. This sphere has radius $R$ and uniform volume charge density $\rho_1$. The two spheres are touching. So, the center of the larger sphere (Sphere 2), which has radius $2R$ and density $\rho_2$, is at a distance of $R + 2R = 3R$ from the origin. Let's place it on the x-axis at $(3R, 0, 0)$.

2. Identifying the Point of Zero Electric Field The problem states that the net electric field is zero at a distance $2R$ from the center of the smaller sphere, along the line joining the centers (the x-axis). There are two such points:

   • Case 1: The point P is at $x = 2R$. This point is between the two centers.
   • Case 2: The point P is at $x = -2R$. This point is on the other side of the smaller sphere.

3. Electric Field Formulas The electric field $\vec{E}$ at a distance $r$ from the center of a uniformly charged non-conducting sphere of radius $a$ and charge density $\rho$ is given by:

   • Outside the sphere ($r \ge a$): $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^3}\vec{r} = \frac{\rho a^3}{3\epsilon_0 r^3} \vec{r}$
   • Inside the sphere ($r < a$): $\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{Qr}{a^3} \frac{\vec{r}}{r} = \frac{\rho}{3\epsilon_0} \vec{r}$ where $\vec{r}$ is the position vector from the center of the sphere to the point in question.

Case 1: Point P at $x = 2R$

1. Position vectors and distances:

   • For Sphere 1 (center at origin): The vector from its center to P is $\vec{r_1} = 2R\hat{i}$. The distance is $r_1 = 2R$. Since $r_1 > R$, the point P is outside Sphere 1.
   • For Sphere 2 (center at $3R\hat{i}$): The vector from its center to P is $\vec{r_2} = 2R\hat{i} - 3R\hat{i} = -R\hat{i}$. The distance is $r_2 = R$. Since $r_2 < 2R$, the point P is inside Sphere 2.

2. Calculating Electric Fields:

   • Electric field due to Sphere 1 at P: $\vec{E}_1 = \frac{\rho_1 R^3}{3\epsilon_0 r_1^3} \vec{r_1} = \frac{\rho_1 R^3}{3\epsilon_0 (2R)^3} (2R\hat{i}) = \frac{\rho_1 R^3}{3\epsilon_0 \cdot 8R^3} (2R\hat{i}) = \frac{\rho_1 R}{12\epsilon_0} \hat{i}$
   • Electric field due to Sphere 2 at P: $\vec{E}_2 = \frac{\rho_2}{3\epsilon_0} \vec{r_2} = \frac{\rho_2}{3\epsilon_0} (-R\hat{i}) = -\frac{\rho_2 R}{3\epsilon_0} \hat{i}$

3. Net Field and Ratio of Densities: The net electric field at P is zero: $\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 = 0$. $\left( \frac{\rho_1 R}{12\epsilon_0} - \frac{\rho_2 R}{3\epsilon_0} \right) \hat{i} = 0$ $\frac{\rho_1}{12} = \frac{\rho_2}{3} \implies \frac{\rho_1}{\rho_2} = \frac{12}{3} = 4$ This corresponds to option D.

Case 2: Point P at $x = -2R$

1. Position vectors and distances:

   • For Sphere 1 (center at origin): The vector from its center to P is $\vec{r_1} = -2R\hat{i}$. The distance is $r_1 = 2R$. Since $r_1 > R$, the point P is outside Sphere 1.
   • For Sphere 2 (center at $3R\hat{i}$): The vector from its center to P is $\vec{r_2} = -2R\hat{i} - 3R\hat{i} = -5R\hat{i}$. The distance is $r_2 = 5R$. Since $r_2 > 2R$, the point P is outside Sphere 2.

2. Calculating Electric Fields:

   • Electric field due to Sphere 1 at P: $\vec{E}_1 = \frac{\rho_1 R^3}{3\epsilon_0 r_1^3} \vec{r_1} = \frac{\rho_1 R^3}{3\epsilon_0 (2R)^3} (-2R\hat{i}) = -\frac{\rho_1 R}{12\epsilon_0} \hat{i}$
   • Electric field due to Sphere 2 at P: $\vec{E}_2 = \frac{\rho_2 (2R)^3}{3\epsilon_0 r_2^3} \vec{r_2} = \frac{\rho_2 (8R^3)}{3\epsilon_0 (5R)^3} (-5R\hat{i}) = \frac{8\rho_2 R^3}{3\epsilon_0 \cdot 125R^3} (-5R\hat{i}) = -\frac{40\rho_2 R}{375\epsilon_0} \hat{i} = -\frac{8\rho_2 R}{75\epsilon_0} \hat{i}$

3. Net Field and Ratio of Densities: The net electric field at P is zero: $\vec{E}_{net} = \vec{E}_1 + \vec{E}_2 = 0$. $\left( -\frac{\rho_1 R}{12\epsilon_0} - \frac{8\rho_2 R}{75\epsilon_0} \right) \hat{i} = 0$ $-\frac{\rho_1}{12} = \frac{8\rho_2}{75}$ $\frac{\rho_1}{\rho_2} = -\frac{12 \times 8}{75} = -\frac{96}{75} = -\frac{32}{25}$ This corresponds to option B.

Conclusion

The problem statement allows for two possible locations for the point of zero electric field, leading to two different possible values for the ratio ${{{\rho _1}} \over {{\rho _2}}}$: 4 and $- {{32} \over {25}}$. Both of these values are present in the options (D and B). Since the question asks for a ratio that can be the answer and this is a single-choice question, we select the one available in the options. Option B is $- {{32} \over {25}}$.

Final Answer is based on Case 2.