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Physics LibreTexts

16.7: Sample problems and solutions

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Exercise 16.7.1

Consider three charged rods of length L which are arranged to form a triangle, as shown in Figure 16.7.1. If the charge on each rod is evenly distributed, what is the net electric field at the center of the triangle?

clipboard_e9837844bb07ad07c50f793c5610dcd3f.png
Figure 16.7.1: A triangle made up of charged rods.
Answer

We can model the object as the sum of three finite length wires of the length, L. In Example 16.3.3, we determined that the electric field produced by a finite wire is:

E=2kλRsinθ0

We can determine geometrically that θ0=π6, as in Figure 16.7.2. The distance, R:

R=L2sinθ0

clipboard_e4c00cb936f970c0ac3edb2bf0c634871.png
Figure 16.7.2

Thus, the field from one wire is given by:

E=2kλRsin(π6)E=kλR

Given that the charge Q is evenly distributed along the rod of length L, we can rewrite the charge density as QL, which gives:

E=kQRL=kQL36L=6kQ3L2

This is the magnitude of the electric field for each side of the triangle. The two positive wires will produce electric fields whose vertical components cancel. The negative wire will produce a field that points downwards. Summing together the electric field vectors:

E=6kQ3L2(cos(π6)cos(π6))ˆx+6kQ3L2(12sin(π6))ˆyE=12kQ3L2ˆy

Which is the final answer.

Exercise 16.7.2

Suppose a dipole is in an electric field E. Show that the dipole will experience simple harmonic motion if the angle between the dipole vector and the electric field vector is small.

Answer

The only net torque on the dipole is from the force from the electric field:

τ=pEsinθ

where we have inserted a minus sign to indicate that this is a restoring torque, in the opposite direction of increasing angle θ. The net torque is equal to the moment of inertia times the angular acceleration:

pEsinθ=Iαα=pEIsinθpEIθ

where in the last equality, we made the small angle approximation (sinθθ). This has the form for simple harmonic motion:

d2θdt2=ω2θω=pEI


This page titled 16.7: Sample problems and solutions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Ryan D. Martin, Emma Neary, Joshua Rinaldo, and Olivia Woodman via source content that was edited to the style and standards of the LibreTexts platform.

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