To find the voltage (potential difference) between the electrodes of the conical capacitor, you can integrate the electric field along a path between the two electrodes. The voltage $V$ can be expressed as:

$V = -\int_{\mathbf{r}_a}^{\mathbf{r}_b} \mathbf{E} \cdot d\mathbf{l}$

Given that your electric field $\mathbf{E}$ is expressed in spherical coordinates, the voltage can be computed by considering a path in the $r$ direction between the two conical surfaces.

Steps to Calculate the Voltage:

1. Choose a Path of Integration: For simplicity, integrate along a radial line at a fixed angle $\theta$ (either $\theta_+$ or $\theta_-$), since the electric field is uniform with respect to $\theta$.

2. Set Up the Integral: The differential path element $d\mathbf{l}$ can be taken as $dr \hat{r} + r d\phi \hat{\phi} + dz \hat{z}$. For your case, focus on $dr$ only, so $d\mathbf{l} = dr \hat{r}$.

3. Dot Product: Compute the dot product $\mathbf{E} \cdot d\mathbf{l}$: $\mathbf{E} \cdot d\mathbf{l} = E_r dr$

4. Integrate: Set up the integration limits from $r_a$ to $r_b$ and substitute your expression for $E_r$: $V = -\int_{r_a}^{r_b} E_r \, dr$

5. Simplify and Solve: Substitute the expression for $E_r$ you have and perform the integration.

Potential Considerations:

• Variation with $\theta$: If the electric field appears not to vary with $\theta$, double-check the derivation of your electric field to ensure that any dependence on $\theta$ has been appropriately accounted for, particularly in relation to the charge distributions or geometric factors.

• Boundary Conditions: Ensure that the boundary conditions (potential at the electrodes) are set correctly when evaluating the integral.

If you provide the limits of integration or any specific aspects you want to address regarding $\theta$, I can offer a more tailored solution.

Questions for Further Exploration:

1. What are the specific values of $r_a$ and $r_b$?
2. Can you clarify the geometry of the conical capacitor further?
3. Are there any assumptions about the uniformity of charge distribution on the electrodes?
4. How do you interpret the variation with $\theta$ in your context?
5. Do you need help with the physical interpretation of the results?

Tip:

When dealing with integrals in cylindrical or spherical coordinates, always pay attention to how the coordinate transformations affect your limits and the forms of $d\mathbf{l}$ and $\mathbf{E}$.