## PHYS208 Fundamentals of Physics II

### Electric Field on Axis of Split Ring

*This approach is the hardest for visualizing vectors!!!*

### Solution:

#### Step 1: Understand the geometry

This configuration is similar to the ring of charge example presented in class.
However, the right half of the ring is now oppositely charged.

#### Step 2: Span the charge distribution

Since the charge is on a ring,
the angle variable *theta* is a convenient choice.
The entire charge distribution can then be spanned by varying *theta*
from 0 to 2 *pi*,
changing the sign of charge half-way around.
Assuming a uniform charge distribution about the ring,
the linear charge density *lambda* will be *Q / (pi a)*,
where we are assigning the variable *a* to be the radius of the ring.
In terms of the charge density, the infinitesimal charge element will be
*dq* = *lambda* *ds* = *lambda* *a* *d(theta)*.

*phi* is also introduced now to get ready for consideration of components of *d***E**.
Note that *phi* is constant once the point of evaluation is chosen; it is does not vary as the
angle *theta* is varied as the charge distribution is spanned.

#### Step 3: Evaluate the contribution from the infinitesimal charge element

Begin by focusing on the left half of the ring.
The infinitesimal contribution *d***E** will point directly away from the
charge infinitesimal, along the line connecting it and the point of evaluation.

#### Step 4: Exploit symmetry as appropriate

The symmetry is exploited by considering the four conjugate elements shown.

The vertical and axial components of *d***E** from one charge infinitesimal
will always be balanced by the contribution from its conjugate.

Thus, only the horizontal *x*-components of the four *d***E** need be
integrated.
All four quadrants of charge contribute the same horizontal component, so we need only integrate
from 0 to *pi* / 2 and multiply the answer by 4.

It takes some maneuvering to visualize exactly what component of *d***E** is needed,
but the right perspective shows that it is proportional to sin(*theta*) and sin(*phi*).

#### Step 5: Set up the integral

#### Step 6: Solve the integral

#### Step 7: The result!

"http://www.physics.udel.edu/~watson/phys208/quiz3extra2.html"

Last updated Oct. 14, 1997.

Copyright George Watson, Univ. of Delaware, 1997.