Step-by-step Solution

1. Identify the Goal: The problem asks to determine how the electric field at the surface of the nucleus (at r = R) depends on the parameter 'a', given the charge density profile ρ(r).

2. Recall Gauss's Law: For a spherically symmetric charge distribution, Gauss's Law is the most effective tool to find the electric field. Gauss's Law states: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ where $Q_{enc}$ is the total charge enclosed within the Gaussian surface.

3. Apply Gauss's Law to the Nucleus: We want to find the electric field at r = R. Let's choose a spherical Gaussian surface with radius r = R, concentric with the nucleus.

   • Due to spherical symmetry, the electric field $\vec{E}$ must be radial and have the same magnitude at all points on the Gaussian surface. Thus, $\vec{E} = E(R) \hat{r}$.
   • The area element $d\vec{A}$ is also directed radially outward, so $d\vec{A} = dA \hat{r}$.
   • The dot product simplifies: $\vec{E} \cdot d\vec{A} = E(R) dA$.

4. Calculate the Left-Hand Side (LHS) of Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \oint E(R) dA = E(R) \oint dA$ The integral $\oint dA$ is simply the total surface area of the Gaussian sphere, which is $4πR^2$. So, the LHS is: $LHS = E(R) \cdot 4\pi R^2$

5. Calculate the Right-Hand Side (RHS) of Gauss's Law: The term $Q_{enc}$ is the total charge enclosed by the Gaussian surface of radius R. Since the nucleus itself has a radius R, the enclosed charge is the total charge of the nucleus. The problem states that the total nuclear charge is Ze. Therefore, $Q_{enc} = Ze$.

6. Combine and Solve for the Electric Field E(R): Equating the LHS and RHS from Gauss's Law: $E(R) \cdot 4\pi R^2 = \frac{Ze}{\epsilon_0}$ Solving for E(R), we get: $E(R) = \frac{Ze}{4\pi \epsilon_0 R^2}$

7. Analyze the Dependence on 'a': Let's examine the expression for the electric field at the surface: $E(R) = \frac{1}{4\pi \epsilon_0} \frac{Ze}{R^2}$ This expression depends on:

   • Z (the atomic number), a constant for a given nucleus.
   • e (the elementary charge), a fundamental constant.
   • $\epsilon_0$ (the permittivity of free space), a fundamental constant.
   • R (the radius of the nucleus), a constant for a given nucleus.

   The parameter 'a' describes the specific details of how the charge is distributed inside the nucleus. However, for any point outside or on the surface of a spherically symmetric charge distribution, the electric field is determined solely by the total charge enclosed, as if it were a point charge at the center. The expression for E(R) does not contain the parameter 'a'.

8. Conclusion: The electric field at r = R is independent of 'a'. This corresponds to option A.

   Note: While the total charge Ze can be calculated by integrating the charge density ρ(r) over the volume, and this integral would depend on 'a' and 'd' (the maximum density from the graph), the problem gives the total charge as Ze. This means that if 'a' were to change, 'd' would have to change correspondingly to keep the total charge constant. But the electric field at the surface only depends on the total charge Ze, not the specifics of the distribution like 'a' or 'd'.