Calculate the electric potential at the point P on the axis of the annulus shown

Question

Calculate the electric potential at the point P on the axis of the annulus shown below, which has surface charge density ?=??r . Then differentiate your expression (the integral you
evaluated) for the electrical potential to show that the electric field at point P is

PLEASE SHOW ALL WORK FOR INTEGRAL...FULL QUESTION ON IMAGE BELOW

Calculate the electric potential at the point P on the axis of the annulus shown below, which has surface charge density = 0r. Then differentiate your expression (the integral you evaluated) for the electrical potential to show that the electric field at point P is E = 2pik 0Z}[-b/ b2+z2 + a/ a2+z2] + [ln(b+ b2+z2/a + a2+z2)]}

Explanation / Answer

consider a circular plastc disk of radius band has positive surface charge density sigma ? on its upper surface

conside the division of disk into one with smaller radius r and radial width dr

then charge dq = ? dA = ?=?(2pi r dr)

dE due ths elemnt at z units from disk = dq/Ae0 =   Z?2 pi r dr/(4pie0 (z^2 +r^2)^3/2

dE = ?Z//4e0)(2rdr /(z^2+r^2)^3/2

E = int dE = ?Z/4e0 *(Z^2+r^2)^3/2 2r dr

applying the upper and loewr lmts i.f for entire disk

E = ?Z/2e0)( a/sqrt(b^2 +z^2 - b/sqrt(b^2 +z^2) + ln(b+sqrt(b^2+z^2)/a+sqrt(a^2+z^2)

as K=1/4pi e0

E = 2pik ? (( a/sqrt(b^2 +z^2 - b/sqrt(b^2 +z^2) + ln(b+sqrt(b^2+z^2)/a+sqrt(a^2+z^2)

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