Two point masses m each carrying charge -q and +q are attached to the ends of a massless rigid n

step 1 Hi friends, here it is given two point charges each having the mass M charge plus Q and minus Q

Two point masses m each carrying charge -q and +q are...

Two point masses $\mathrm{m}$ each carrying charge $-\mathrm{q}$ and $+\mathrm{q}$ are attached to the ends of a massless rigid non-conducting rod of length $\ell$. The arrangement is placed in a uniform electric field $\mathrm{E}$ such that the rod makes a small angle $5^{\circ}$ with the field direction. The minimum time needed by the rod to align itself along the field is $\ldots \ldots .$ (A) $t=\pi \sqrt{[}(2 M \ell) /(3 q E)]$ (B) $\mathrm{t}=(\pi / 2) \sqrt{[}(\mathrm{M} \ell) /(2 \mathrm{q} \mathrm{E})]$ (D) $t=2 \pi \sqrt{[}(M \ell / E)]$

Two point masses $\mathrm{m}$ each carrying charge $-\mathrm{q}$ and $+\mathrm{q}$ are attached to the ends of a massless rigid non-conducting rod of length $\ell$. The arrangement is placed in a uniform electric field $\mathrm{E}$ such that the rod makes a small angle $5^{\circ}$ with the field direction. The minimum time needed by the rod to align itself along the field is $\ldots \ldots .$
(A) $t=\pi \sqrt{[}(2 M \ell) /(3 q E)]$
(B) $\mathrm{t}=(\pi / 2) \sqrt{[}(\mathrm{M} \ell) /(2 \mathrm{q} \mathrm{E})]$
(D) $t=2 \pi \sqrt{[}(M \ell / E)]$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes the oscillatory motion of a system where the restoring force (or torque in rotational systems) is proportional to the displacement from equilibrium. In the context of a dipole in a uniform electric field with a small angular displacement, the torque leads to angular SHM. Knowledge of SHM allows one to determine parameters such as the period of oscillation, which in turn can be related to the minimum time required for the dipole to align with the field.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to changes in its rotational motion. In the context of a dipole made of two point masses connected by a rigid rod, it is calculated by summing the products of each mass and the square of its distance from the axis of rotation. This concept is important when relating the applied torque to the resulting angular acceleration.

Small Angle Approximation

When the angular displacement of a system is small, trigonometric functions such as sine can be approximated by their argument (in radians). This approximation simplifies the analysis of rotational motion by linearizing the otherwise nonlinear equations. It is particularly useful in problems involving oscillations or small deviations from equilibrium, where the resulting equations are of the simple harmonic motion form.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a fixed distance. The dipole is characterized by a dipole moment, which is a vector quantity defined as the product of one of the charges and the distance between the charges. This concept is central when analyzing the behavior of such a system in an external electric field, where the forces acting on the charges are not necessarily equal and opposite, leading to a net torque on the dipole.

Dipole Torque

The net torque acting on an electric dipole placed in a uniform external electric field is given by the cross product of the dipole moment and the electric field. This torque tends to align the dipole with the electric field. Understanding how this torque is derived and how it influences the rotational motion of the dipole is crucial for solving problems related to the orientation dynamics of dipoles in fields.

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