PHYS208 Fundamentals of Physics II

Electric Field in the Plane of a Split Ring

This approach has the toughest trigonometry!!!

[split ring dipole]


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.

[split ring dipole]

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 dE.

Step 3: Evaluate the contribution from the infinitesimal charge element

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

[split ring dipole]

[split ring dipole]

Please note that the law of cosines was used to represent r2 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.

[split ring dipole]

The vertical components of dE from one charge infinitesimal will always be balanced by the contribution from its conjugate.

Thus, only the horizontal x-components of the two dE 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.

[split ring dipole]

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

[split ring dipole]

Step 6: Solve the integral

[split ring dipole]

Step 7: The result!

[split ring dipole]


"http://www.physics.udel.edu/~watson/phys208/quiz3extra2.html"
Last updated Oct. 26, 1997.
Copyright George Watson, Univ. of Delaware, 1997.