How can I find the location where the electric field is an extremum (maximum or

Question

How can I find the location where the electric field is an extremum (maximum or minimum). In terms of a.

Question 1: (5 points) The electric field, for a uniformly charged ring of radius a.with total charge Q, at a point on the axis (i.e. perpendicular to the plane of the ring) is given by measured along the axis with 0 being at the center of the ring. The variation of the above function is kO- where z is shown below. -2-1 4 z/a Find the location where the electric field is an extremum (maximum or minimum). Your answer will be in terms of a.

Explanation / Answer

Electric intensity at a distance z from the center of the ring on its axis is given by
E = [Q/(4o)] z / (z^2 + a^2)^(3/2)
=> E is maximum where the value of
f (z) = z / (z^2 + a^2)^(3/2) is maximum
For f (z) to be maximum, f '(z) = 0 and f"(z) < 0
ln [f (z)] = lnz - (3/2) ln(z^2 + a^2)
=> f '(z) / f (z) = 1/z - 3z / (z^2 + a^2)
f '(x) = 0
=> 1/z - 3z / (z^2 + a^2) = 0
=> (z^2 + a^2) - 3z^2 = 0
=> z^2 = a^2/2
=> z = ± a/2
=> E has extremum for z = ± a/2

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.