Question

A Two parallel rings, each of radius R, are separated by a distance R. A positive charge^+Q is uniformly distributed around the upper ring and a negative charge^-Q is uniformly distributed around the lower ring. Let z be the vertical coordinate, with z = 0 taken to be the center of the lower negatively charged ring. What is the direction and magnitude of the electric field at the point A on the vertical axis, a distance z = R/2 from the center of the lower ring? What is the potential difference, V(P)-V(infinity), between a point P located on the vertical axis a distance^+z from the center of the lower ring, and infinity? Set^(infinity) = 0.

Please show all work!

Answer 1

( 200 г р 4.0 First you need to understand What is the potential at a point P, lying on the anis of a uniformly Changed ning of radius R. lets concider an elemental ungth de on which elemental Change der=xoe, where als the line change density a.it a is the total change on the ning then, as a man so potential Change da from the is, Cassuming due to this elemental at a distance z center of the ning potential is o at infinity) - dvs. 4nt. R2+72 Ant. R2+22 Ant. J R² +22

ardo 4tto J R2+22 AVE 4at. Jerz2 =) Ava a aar AntoTR2 +22 » Av= 2 to JP 122 of electric Now, the zr component field, a ďT dv Ezs az Art. azu VR2+zL . 9822-ci me venova) q z Ez = Ant. (R² +223/2 e due to symmetry all the other components ie, Enafu=0

a ) +7 A 720 si, magnitude of electric field at point Acie, a= 2 ) is E = Q (2) Anto CRZA R233/2 * Ano, y me he QR Ant. R² & 3 - E = Ahto et son 40t. R² 5312 >> E = 20 20 520E.R? So E CER?

& direction will be downward ie alony _'ve zranis. Now, the upper Potential sing due to point z at iy . Arto JR²+ (2-R2 the lower tue to & potential ning is . ve z into TRZ difference with v(8)=0 1. So, potential total I ove utva nova et. [vanhen ✓ R2+z2 Anto