Question

Show the derivation steps between (22-13) to (22-16) please include descriptions of properties/laws followed. Thank you.

lect all the perpendiculal t Adding Components. We have another omponents are in the positive direction of the z axis, so we can just add p as scalars. Thus we can already tell the direction of the net el the : directly away from the ring. From Fig. 22-12, we see that the paralled a onents each have magnitudedE cos 6, but θ is another illegal symbol el con. eplace cos o with legal symbols by again using the right triangle in Fig write (22-13) ultiplying Eq. 22-12 by Eq. 22-13 gives us the parallel field com ch charge element: ds. cos θ . (22-14) 4π80 (z2 + R2)32 Integrating. Because we must sum a huge number of these components all, we set up an integral that moves along the ring, from element to ele m a starting point (call it s 0) through the full circumference (s 2TR) quantity s varies as we go through the elements; the other symbols in Eq. 22-14 ain the same, so we move them outside the integral. We find (22-15) is afine answer, but we can also switch to the total cdharge by using A-g/2aR) 4z mlR (charged ring). (22-16) charge on the ring is negative, instead of positive as we have assumed,the itude of the field at P is still given by Eq. 22-16. However, the electric r then points toward the ring instead of away from it. et us check Eq. 22-16 for a point on the central axis that is so far away For such a point, the expression z2+ R2 in Eq. 22-16 can be appr and Fa 3.16 bag approximate

Answer 1

This question is based on the following concepts/laws/properties

1. Properties of cosine

2. Pythagoras Theorem

3. Electric field due to a charge

4. Charge density concept

5. Concept of integration

6. Mathematical simplification.

The detailed solution is described below

THANKS!!!