Question

Under what condition does a particle experiencing electric force perform simple harmonic motion?

Answer 1

The equation describing Simple Harmonic Motion contains complete description of the motion, and is represented as

`y(t)=Asinomegat` ....(1)
where `y` is position of particle in the `y`-axis, `A` is maximum amplitude `y` can achieve and `omega` is angular frequency of the particle executing SHM and is a system constant.

It acceleration is given as

`|veca|=ddoty(t)=-omega^2Asin omegat` .....(2)

Now a particle of mass `m` having charge `q` moving in an electric field experiences force as

`vecF_e=qvecE`

From Newton's second Law of motion, acceleration of the particle is given as

`veca=q/mvecE` ......(3)

For the charge particle to execute SHM equation (2) and (3) must match. Mass of the particle being constant, we have

`q|vecE|=-ksin omegat` .....(4)
where `k` is an other constant dependent of system parameters.

Equation (4) lays down conditions for the particle to execute SHM.

Needless to say that the particle cannot execute SHM if both charge and electric field are constants.