Question

A small electrically charged bead with the mass m and charge Q can slide on a circular insulating string without friction. The radius of the circle is r. A point-like electric dipole is at the center of the circle with the dipole moment P lying in the plane of the circle. Initially the bead is at an angle θ = 플 +6, where δ is infinitely small, as shown schematically in the figure below. Figure 3: Bead on a string (a) How does the bead move after it is released? Find the bead velocity as a function of the angle θ. (b) Find the normal force exerted by the string on the bead.

Answer 1

(a) Find the bead velocity as a function of the angle, $\theta$.

$\Rightarrow$ Applying the law of conservation of energy for a bead of mass & charge in the field of dipole with dipole moment.

Then, we have

(1/2) m v² + Q P cos $\theta$ / r² = (1/2) m v₀² + Q P cos ($\pi$/2) / r²

where, v₀ = intial velocity of bead = 0

(1/2) m v² + Q P cos $\theta$ / r² = 0 + 0

(1/2) m v² = - Q P cos $\theta$ / r²

v² = - (2 Q P cos $\theta$) / m r²

v = _$\sqrt{}$- (2 Q P cos $\theta$) / m r²

How does the bead move after it is released?

$\Rightarrow$ The bead moves along a circular path until it reaches the point opposite its starting position. The bead stops there and then goes back executing a periodic motion.

(b) Find the normal force exerted by the string on bead.

$\Rightarrow$ The radial component of the force on charge due to the dipole can be calculated as the derivative of an electric potential energy with respect to 'r' which given as -

F_{r $\rightarrow$dipole} = d[(Q P cos $\theta$) / r²] / dr

F_{r $\rightarrow$dipole} = - (2 Q P cos $\theta$) / r³

For a circular motion, we have

F_{r $\rightarrow$centripetal} = F_{r $\rightarrow$dipole} + F_{r $\rightarrow$string}

m v² / r = - (2 Q P cos $\theta$) / r³ + F_{r $\rightarrow$string}

- (m / r) [(2 Q P cos $\theta$) / m r²] = - (2 Q P cos $\theta$) / r³ + F_{r $\rightarrow$string}

- (2 Q P cos $\theta$) / r³ = - (2 Q P cos $\theta$) / r³ + F_{r $\rightarrow$string}

F_{r $\rightarrow$string} = - [(2 Q P cos $\theta$) / r³] + [(2 Q P cos $\theta$) / r³]

F_{r $\rightarrow$string} = 0