Suppose a capacitor consists of two coaxial thin cylindrical conductors. The inner cylinder of radius ra has a charge of +Q, while the outer cylinder of radius rb has charge -Q. The electric field E at a radial distance r from the central axis is given by the function: E = αe-r/a0 + β/r + b0 where alpha (α), beta (β), a0 and b0 are constants. Find an expression for its capacitance. First, let us derive the potential difference Vab between the two conductors. The potential difference is related to the electric field by: V subscript a b end subscript equals integral subscript r subscript a end subscript superscript r subscript b end superscript E d r equals negative integral subscript r subscript b end subscript superscript r subscript a end superscript E d r Calculating the antiderivative or indefinite integral , Vab = (-αa0e-r/a0 + β + b0 )right enclose blank end enclose subscript space r subscript a end subscript superscript space r subscript b end superscript By definition, the capacitance C is related to the charge and potential difference by: C = / Evaluating with the upper and lower limits of integration for Vab, then simplifying: C = Q / ( (e-rb/a0 - e-ra/a0) + β ln( ) + b0 ( ) )
Suppose a capacitor consists of two coaxial thin cylindrical conductors. The inner cylinder of radius ra has a charge of +Q, while the outer cylinder of radius rb has charge -Q. The electric field E at a radial distance r from the central axis is given by the function:
E = αe-r/a0 + β/r + b0
where alpha (α), beta (β), a0 and b0 are constants. Find an expression for its capacitance.
First, let us derive the potential difference Vab between the two conductors. The potential difference is related to the electric field by:
V subscript a b end subscript equals integral subscript r subscript a end subscript superscript r subscript b end superscript E d r equals negative integral subscript r subscript b end subscript superscript r subscript a end superscript E d r
Calculating the antiderivative or indefinite integral ,
Vab = (-αa0e-r/a0 + β
+ b0
)right enclose blank end enclose subscript space r subscript a end subscript superscript space r subscript b end superscript
By definition, the capacitance C is related to the charge and potential difference by:
C =
/
Evaluating with the upper and lower limits of integration for Vab, then simplifying:
C = Q / (
(e-rb/a0 - e-ra/a0) + β ln(
) + b0 (
) )
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