What Is the Optimal Inner Radius to Minimize Electric Field in a Coaxial Cable?

[jesuslovesu]

Homework Statement

If anyone could clarify this statement for me, I'm having a bit of difficulty interpreting what the heck I'm supposed to do:

"For a given potential difference V0 between the inner and outer conductors and for a given fixed value of b, determine the inner radius a for which the largest value of the electric field is a minimum."I found the potential fairly easily using Laplace's equation
$$V(r) = \frac{V_0 ln(r/b)}{ln(a/b)}$$ (a is inner radius, b is outer)
I know that the electric field is the negative gradient of potential, but I really don't know what they are getting at.

 

[Nate810]

Homework EquationsV(r) = \frac{V_0 ln(r/b)}{ln(a/b)}E=-\nabla VThe Attempt at a SolutionI think what they are looking for is the minimum value of the largest electric field (so the maximum of the minimums). You can use the equation for the electric field to determine the electric field at various points on the surface of the inner and outer conductors. Then, you can compare the values at these points to find the point with the maximum of the minimums. This will give you the inner radius a that produces the largest value of the electric field that is also a minimum.

 

[Moetasim]

Based on the information provided, it seems like you are being asked to determine the inner radius a that will result in the minimum electric field for a given potential difference and fixed value of b. In other words, you are trying to find the optimal size for the inner conductor that will minimize the electric field.

To solve this, you can start by taking the negative gradient of the potential equation you found. This will give you the expression for the electric field as a function of radius. Then, you can set this expression equal to zero and solve for the inner radius a. This will give you the value of a that results in the minimum electric field.

Alternatively, you can plot the electric field as a function of radius for different values of a and visually determine the value of a that results in the minimum electric field.

Overall, it seems like this problem is asking you to optimize the design of a Laplacian coaxial cable to minimize the electric field. This could have practical applications in reducing the potential for electrical breakdown or interference in the cable.