- #1

gruba

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## Homework Statement

Three

__conductors, with radius__

**very long (theoretically infinite long) hollow cylindrical***are in vacuum. Inner and central conductor are charged, and outer conductor is grounded. Potentials of inner and central conductors with reference point relative to outer conductor are*

**a,b,c (c>b>a)***. Find*

**V**_{a},V_{b}__of all three conductors.__

**longitudinal charge density**## Homework Equations

__of cylindrical conductor can be derived using Gauss law for vacuum:__

**Electric field***where*

**E=Q**^{'}/(2πrε_{0}),*is longitudinal charge density.*

**Q'****of cylindrical conductor is given by:**

__Electric potential__*, where*

**V=∫Edl***represents integration by radius.*

**dl**## The Attempt at a Solution

If outer conductor is grounded, and it is a referent point to potentials

*and*

**V**_{a}*integration for*

**V**_{b},*will be from*

**V**_{a}*to*

**(a***to*

**b)+(b**

**c),**

VV

_{a}=(Q^{'}/(2πε_{0}))*(ln(b/a)+ln(c/b))Integration for

*will be from*

**V**_{b}*to*

**(b**

**c),**

VV

_{b}=(Q^{'}/(2πε_{0}))*ln(c/b)We need to find longitudinal charge density for each capacitor, so for first we derive it from

**V**

Q_{a}:Q

^{'}=(2πε_{0}V_{a})/(ln(b/a)+ln(c/b))For second conductor, we derive it from

**V**

Q_{b}:Q

_{'}=(2πε_{0}V_{b})/(ln(c/b))Third conductor is grounded, so the potential of the third conductor is equal to zero, thus the longitudinal charge density of the third conductor is equal to zero.

Could someone check this, and help if something is not correct?

Thanks for replies.