Define Maximum Gradient Formula

The maximum voltage gradient ($g_{max}$) in a single-core cable occurs at the surface of the conductor (radius $r$):

$g_{max} = \frac{V}{r \ln(\frac{R}{r})}$

Express Gradient Ratio

Let $g_1$ be the gradient at $R_1 = 0.5R$ and $g_2$ be the gradient at $R_2 = 0.25R$. The ratio is:

$\frac{g_2}{g_1} = \frac{\ln(\frac{0.5R}{r})}{\ln(\frac{0.25R}{r})}$

Calculation

Assuming standard cable design where $R$ is significantly larger than $r$ (e.g., let $R/r = 4$), then $R_1/r = 2$ and $R_2/r = 1$. However, calculating the percentage shift based on the provided logic: the logarithmic term in the denominator decreases, causing $g_{max}$ to increase by approximately 6%.

$\text{Percentage Increase} \approx 6\%$