Section 3.3 · Home exercises · 8 of 9
Home exercise 3.3.8: How much voltage can it take?
How much voltage can it take?
#Two concentric spherical conductors have radii and . The outer conductor is grounded and the inner one is held at the potential . The space between them is air, whose dielectric strength is . Find the largest voltage that can be applied without breakdown.
Hint: use the solution of Problem 3.3.3 and find where the field is largest.
Take the field from the Laplace solution of Problem 3.3.3. Breakdown starts where the field is largest, so we only need the maximum of in the gap and the condition .
From the Laplace solution of Problem 3.3.3 with and ,
The field decreases as , so it is largest at the surface of the inner conductor, . Breakdown is avoided as long as this maximum does not exceed the dielectric strength:
With ,
exactly the dielectric strength, as intended. At the outer sphere the field is only of this value.
For a fixed outer radius , is largest when , i.e. for , giving here. A very thin inner conductor concentrates the field, while a very thick one leaves too small a gap. Exercise 3.3.10 finds the corresponding optimum for a coaxial line (Figure 3.3.28).