Electric charges and field Part – 5
5. Oscillation of a charge: Two +q charge particles are placed at point A (0, a) and B (0, -a) on the Y axis and another charge particle –Q is released from rest at point C (b,0) on the X axis. At any instant -Q is at point D of distance x from origin.
*(i) The net force on -Q due to charges at A and B is F/ = 2F cos 
= – 
As the force is nonlinear then the motion of charge Q is oscillatory. Amplitude of the charge Q is b.
(ii) If point C is very closed to the origin and a 
x then, F/ = – 
= – 
(using binomial theorem).
So, the motion of charge -Q is simple harmonic with angular frequency ω = 
.
(iii) Position of the charge Q where force of attraction is maximum or minimum: 
= 0
Or, – 
= 0
Or, 
= 0
Or, 3x2 = a2 + x2
Or, 2x2 = a2
So, x = 
.
6. Relation between dielectric constant and density of medium: Two identical balls, each of equal mass m, density ρ and charge q are suspended from a common point by two insulating strings of same length making angle 2θ. Now, both the balls are immersed in a liquid of dielectric constant k and the angle between the strings remain unchanged.
The density of the liquid is σ.
When the charges are in air, T cos = Mg and T sin = F
where T = tension on the string and F = force between the charges.
So, F = Mg tan = Vρg tan ——-(i) where V = volume of each charge.
When the charges are in liquid, T/ cos = (Mg – U) and T/ sin = F/ where T/ = tension on the string U = buoyant force and F/ = force between the charges.
So, F/ = (Mg – U)tan = (Vρg – Vσg) tan ——–(ii)
From, equation (i) and equation (ii) we get, 
= 
Again F/ = 
so, k = 
.
7. Force between two mutually perpendicular charged wires: A uniformly charged wire of length l and linear charge density λ1 is placed perpendicularly with infinitely long charged wire of linear charge density λ2 with a separation r as shown in figure.
The electric field at a distance x from the infinitely long wire is E = 
We consider an elementary length dx of the wire 1 having charge dq = λ1dx.
The force acting on the elementary part is dF = Edq = λ1λ2 
The force acting on wire 1 due to infinitely long wire is
F = 
= λ1λ2 
= 
In 
= In 
.
8. Two point charges each of charge q are separated by a distance 2d. An electron of charge e describes a circular path of radius r due to the attraction of the charges in a plane, bisecting perpendicularly the line joining the two point charges. Determine the orbital speed of the electron.
The force of attraction between q and e is F = 
.
The component of forces acting towards the centre of the circular path is
F/ = 2F sin = 
This force provides the centripetal force to the electron.
If v is the speed of electron, then, = 
v = 
.
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