Why does Kirchhoff's law give wrong voltage between points in a point charge field?

15ongm · Apr 2, 2016

Homework Statement

Cross-section diagram of a neutral spherical conductor of radius R with a spherical cavity of radius R/2 above its center, containing a negative uniform spherical shell of charge -Q and radius R/4; labelled points a, b, c, d and e, with e at distance 5R/4 from c.

$Printed problem 4.2 Voltages (15 pts) with a hint and note; the note gives V_{b←a} = k_C q (1/r_b − 1/r_a) for the voltage from a point charge q.$

Two equations written as V subscript e-from-d equals and V subscript e-from-a equals, with the right-hand sides not shown.

Homework Equations

The voltage between any two points due to the field from a point charge q is:

Equation for the voltage from point a to point b due to a point charge q: V(b←a) = k_C q (1/r_b − 1/r_a).

Kirchoff's voltage laws, which states that the total voltage around a closed loop must be zero, i.e.:

$Equation: V_ba = V_{b<-a} = V(b) - V(a) = minus the integral from a to b of E(r) dot dr, with vector symbols on E and r.$

The Attempt at a Solution

To find V(e<-d) I used Kirchoff's Voltage Law:

Handwritten derivation: 0 = V(d<-a) + V(c<-e) + V(e<-d), then 0 - V(e<-c), then 0 = 4k_cQ/(5R) + V(e<-d), giving V(e<-d) = -4k_cQ/(5R).

To find V(e<-a) I used the fact that you can break voltages into parts:

Handwritten derivation of V(e<-a) as the sum of V(e<-c) and V(c<-a): -k_cQ/(5R/4) plus -2k_cQ/R, simplified to -14k_cQ/(5R) times a charge term.

But I know my answers are wrong b/c this is what the solutions say:

My question is, what is wrong with the way I'm calculating V(e<-d) and V(e<-a)?

mfb · Apr 3, 2016

How did you get the expression for V_c<-e?
The detour via c is unnecessary.

The expression for V_c<-a is wrong as well, and I guess it is wrong for the same reason.

Why does Kirchhoff's law give wrong voltage between points in a point charge field?

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

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