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 λ₂with 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 = .