A dust particle with mass of 0.050 g and a charge of 2.0 x 10⁻⁶ C is in a region of space where the potential is given by V(x) = (2.0 V/m²) x² - (3.0 V/m³)x³. If the particle starts at x = 2.5m, what is the initial acceleration of the charge?

검증된 단계별 안내

Convert the mass of the dust particle from grams to kilograms, as the SI unit for mass is kilograms. Use the conversion factor: 1 g = 0.001 kg. Thus, the mass is m = 0.050 g × 0.001 kg/g.

Determine the electric field E(x) from the given potential function V(x). The electric field is the negative gradient of the potential: E(x) = -dV/dx. Differentiate V(x) = (2.0 V/m²)x² - (3.0 V/m³)x³ with respect to x.

Evaluate the electric field E(x) at the given position x = 2.5 m. Substitute x = 2.5 m into the expression for E(x) obtained in the previous step.

Calculate the force acting on the charged particle using the formula F = qE, where q is the charge of the particle (2.0 × 10⁻⁶ C) and E is the electric field at x = 2.5 m.

Determine the initial acceleration of the particle using Newton's second law, a = F/m, where F is the force calculated in the previous step and m is the mass of the particle in kilograms.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Electric Potential and Electric Field

Electric potential (V) is a measure of the potential energy per unit charge at a point in an electric field. The electric field (E) can be derived from the potential by taking the negative gradient, E = -dV/dx. In this case, the potential function V(x) is given, and understanding how to calculate the electric field from it is crucial for determining the force acting on the charged particle.

Force on a Charged Particle

The force (F) acting on a charged particle in an electric field is given by F = qE, where q is the charge of the particle and E is the electric field strength. This relationship is fundamental in physics as it allows us to calculate the force experienced by the particle due to the electric field, which is essential for finding its acceleration.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration (a) of an object is directly proportional to the net force (F) acting on it and inversely proportional to its mass (m), expressed as F = ma. This principle is vital for calculating the acceleration of the dust particle once the force acting on it has been determined from the electric field.