## PHYS208 Fundamentals of Physics II

### Electric Field in the Plane of a Split Ring

*This approach has the toughest trigonometry!!!*

### Solution:

#### Step 1: Understand the geometry

This configuration is similar to that in Quiz 3.
However, we are now considering a point outside the ring, rather than in the center.
The point of evaluation is still in the plane of the ring, however.

#### 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**.

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

Begin by focusing on the right 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.

Please note that the law of cosines was used to represent *r*^{2} in terms of
*a*, *d*, and importantly, *theta*, our variable of integration.

#### Step 4: Exploit symmetry as appropriate

The symmetry is exploited by considering the conjugate element shown.

The vertical 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 two *d***E** need be
integrated.
Both halves contribute the same horizontal component, so we need only integrate
from 0 to *pi* and multiply the answer by 2.

Please note that the law of sines was used to represent the sine of *phi* in terms of
*a*, *d*, and the sine of *theta*.

#### 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. 26, 1997.

Copyright George Watson, Univ. of Delaware, 1997.