Two particles are in a uniform electric field that points in...

Two particles are in a uniform electric field that points in the $+x$ direction and has a magnitude of $2500 \mathrm{N} / \mathrm{C}$. The mass and charge of particle 1 are $m_{1}=1.4 \times 10^{-5} \mathrm{kg}$ and $q_{1}=-7.0 \mu \mathrm{C},$ while the corresponding values for particle 2 are $m_{2}=2.6 \times 10^{-5} \mathrm{kg}$ and $q_{2}=+18 \mu \mathrm{C} .$ Initially the particles are at rest. The particles are both located on the same electric field line but are separated from each other by a distance $d$. Particle 1 is located to the left of particle $2 .$ When released, they accelerate but always remain at this same distance from each other. Find $d$.

Key Concepts

Equating Accelerations for Constant Separation

When two objects maintain a constant distance from each other while accelerating, it implies that they experience the same acceleration. This concept is used to set up conditions that balance forces from both the external field and mutual interactions (such as those described by Coulomb's Law) so that the relative motion between the objects is null.

Coulomb’s Law

Coulomb’s Law describes the electric force between two point charges, stating that the magnitude of the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is critical when considering interactions between charged particles in addition to external fields.

Uniform Electric Field

A uniform electric field is one in which the electric field strength (both magnitude and direction) remains constant throughout the region. This concept is essential in analyzing the forces acting on charges within such a region, as each charge experiences the same field irrespective of its position.

Electric Force on a Charge

The electric force on a charged particle in an electric field is given by the equation F = qE, where q is the charge and E is the electric field. This concept determines how the field influences the motion of charged particles, leading to acceleration according to the sign and magnitude of the charge.

Newton’s Second Law

Newton’s Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This principle is used to relate the forces acting on charged particles to their resulting accelerations, enabling the analysis of their motion.

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