Electromagnetism - Coulomb's Law



Two point charges with the same charge +Q are fixed along the x-axis and are a distance 2R apart as shown. A small particle with mass m and charge –q is placed at the midpoint between them. What is the angular frequency ω of small oscillations of this particle along the y-direction? 

A. 

B.

C.

D.

E.
(GR9277 #65)

Solution:

Check the unit, ω → sec−1

with
Q = q → Coulomb
R = meter
m = kg
ϵ0 = coulomb2 / (newton · meter2) = coulomb· sec2/ (kg · meter3)

A. coulomb2 · kg · meter/  (coulomb2 · sec2 · kg · meter2) =  meter/sec2
→ FALSE.

B. Identical unit with A. → FALSE

C. 1/sec2 → FALSE

D. √m/sec → FALSE

E. 1/sec → TRUE

Answer: E


Calculation

Coloumb's Law: 

In this case,



(Electric fields from both Qs are in x-and y-direction, but in x-axis, they cancel each other.)

For small θ, sin θ ≈ tan θ = y/R

Hooke's Law in y-axis: 

Angular frequency,  

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