A proton is traveling horizontally to the right at $4.50 \times 10^{6}$ m/s. Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of $3.20$ cm.

검증된 단계별 안내

First, identify the initial velocity $ v_i $ of the proton, which is given as $ 4.50 \times 10^6 $ m/s, and the final velocity $ v_f $ which is 0 m/s since the proton comes to rest.

Next, note the distance $ d $ over which the proton comes to rest, which is given as 3.20 cm. Convert this distance to meters by dividing by 100, resulting in $ 0.032 $ m.

Use the kinematic equation $ v_f^2 = v_i^2 + 2a d $ to solve for the acceleration $ a $. Rearrange the equation to $ a = \frac{v_f^2 - v_i^2}{2d} $. Substitute the known values to find $ a $.

Recall that the force $ F $ acting on the proton is related to the electric field $ E $ by the equation $ F = qE $, where $ q $ is the charge of the proton ($ 1.60 \times 10^{-19} $ C). Also, $ F = ma $, where $ m $ is the mass of the proton ($ 1.67 \times 10^{-27} $ kg).

Combine the equations $ F = ma $ and $ F = qE $ to solve for the electric field $ E $. Rearrange to $ E = \frac{ma}{q} $. Substitute the values for $ m $, $ a $, and $ q $ to find the magnitude of the electric field. The direction of the electric field will be opposite to the direction of the proton's initial velocity to bring it to rest.

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주요 개념

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

Electric Field

An electric field is a region around a charged particle where a force would be exerted on other charged particles. It is represented by the vector E and measured in newtons per coulomb (N/C). The direction of the electric field is the direction of the force it would exert on a positive charge.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It involves equations that relate displacement, velocity, acceleration, and time. In this problem, kinematics helps determine how the proton's velocity changes as it travels through the electric field.

Work-Energy Principle

The work-energy principle states that the work done by forces on an object results in a change in its kinetic energy. For a proton brought to rest, the work done by the electric field equals the initial kinetic energy of the proton. This principle helps calculate the required electric field strength to stop the proton over a given distance.