What is the Period of Oscillation for a Charge in an Electrostatic Field?

[triplebig]

Hey guys I'm from Brazil and I'm studying physics to pass in one of the best universities here. However, the physics test is much harder than what we study at school. I am currently working on a book called "Selected Elementary Physics", by MIR Moscou, and I am having a lot of trouble. I don't have anyone who can help me solve these problems, and I am hoping these forums would be a good solution to my problem. I'm from Brazil and my english terminology on Physics is very bad, and I might make bad translations, which I will point out by (?)(?). I'll start with a first one, hoping that it somebody can solve.

Homework Statement

Two equal (?)"Charges"(?) $$+Q$$ are fixed and located at a distance $$a$$ from each other. Along the symmetry axis of these charges, a third charge, $$-q$$, can be moved, which has a mass $$m$$. Considering the distance from the $$-q$$ particle to the line that unites the $$+Q$$ charges, determine the (?)oscillations period(?) of the $$-q$$ charge.

Homework Equations

$$F\,=\,\frac{1}{4\pi\epsilon}\,.\,\frac{Q\,.\,q}{d^2}$$

The Attempt at a Solution

Since it says the distance is small, I assumed it to be infinetely small, and considered the distance from $$+Q$$ to $$-q$$ to be $$a$$ as well. I then used $$F\,=\,m.a$$, with no effect. I don't know how to approach this problem, any help is appreciated.

Answer: $$\Large{T\,=\,\pia\,\sqrt{\frac{\pi\epsilon _o.m.a}{Qq}}\,.\pi\,.\,a}$$

 

[Shooting Star]

The Attempt at a Solution

Since it says the distance is small, I assumed it to be infinetely small, and considered the distance from $$+Q$$ to $$-q$$ to be $$a$$ as well. I then used $$F\,=\,m.a$$, with no effect. I don't know how to approach this problem, any help is appreciated.

(Pl don't use same symbols for different quantities, like you have used 'a'.)

The distance by which the -q is disturbed is small, and you have considered it to be infinitesimally small, which is all right. From there, how did you jump to equating it to 'a'? Is ‘a’ infinitesimally small?

Draw a diagram. The two +Qs are at A and B. The –q charge is at the midpoint C of AB. Suppose it is displaced to D, where D is on one of the perpendicular bisectors of the segment AB, and CD is very small. Let CD=x and ∟CAD= θ.

Now calculate the vertical, i.e., the force along DC on –q. You get the force equation by equating $m(d^2x/dt^2)$ to that force. Since θ is very small, sin θ ~ θ, and BD ~ BC.

Now you will land up with a familiar equation. You know how to find the time period of such an oscillation.

 

[Moetasim]

Hello,

Thank you for sharing your problem with us. It seems like you are trying to find the period of oscillation for the -q charge in the presence of the two fixed +Q charges. This is a problem involving electrostatic forces and motion, which can be solved using the principles of classical mechanics.

First, we can start by considering the forces acting on the -q charge. There are two forces at play here: the electrostatic force from the two +Q charges and the force of gravity. The electrostatic force can be calculated using Coulomb's law, as shown in the equation you provided. The force of gravity, on the other hand, is given by F = mg, where m is the mass of the -q charge and g is the acceleration due to gravity.

Next, we can use Newton's second law, F = ma, to set up an equation of motion for the -q charge. Since the charge is moving along the symmetry axis of the two +Q charges, we can assume that the motion is one-dimensional. This means that we can represent the displacement of the -q charge as x, and the acceleration as a.

Now, we can combine the equations for the electrostatic force and the force of gravity to get an equation for the acceleration of the -q charge: a = (1/4πε) (Qq/x^2) - g. Since we are dealing with small displacements (as you assumed), we can use the small angle approximation to simplify this equation to a = -(1/4πε) (Qq/a^2) - g. This is a simple harmonic motion equation, with the restoring force being the electrostatic force from the two +Q charges.

Using the equation for simple harmonic motion, we know that the period of oscillation is given by T = 2π√(m/k), where m is the mass of the -q charge and k is the spring constant. In this case, k is equal to (1/4πε) (Qq/a^2), so we can rewrite the equation for the period as T = 2π√(m/[(1/4πε) (Qq/a^2)]).

Simplifying this equation, we get T = 2π√(4πεm/(Qq/a^2)). Rearranging, we get T = 2π√(4πε