CALC A uniform electric field’s strength is increasing with time as E=(1.5×10^4+(5.0×10^10 s^−1)t) N/C . A proton is released in the field from rest at t=0 . What is the proton’s speed 1.0 μs later?

Verified step by step guidance

1. Identify the given values: Electric field strength E as a function of time t is given by E = (1.5 \times 10^4 + 5.0 \times 10^{10} t) N/C, where t is in seconds. The charge of a proton (q) is approximately 1.602 \times 10^{-19} Coulombs, and the mass of a proton (m) is approximately 1.672 \times 10^{-27} kg.
2. Calculate the force exerted on the proton by the electric field at any time t using the formula F = qE, where F is the force, q is the charge of the proton, and E is the electric field strength.
3. Determine the acceleration (a) of the proton using Newton's second law, F = ma, where m is the mass of the proton. Rearrange the formula to find a = F/m.
4. Integrate the acceleration over time to find the velocity of the proton as a function of time. Since the proton starts from rest, the initial velocity v_0 = 0. The velocity v at any time t can be found by integrating the acceleration: v(t) = \int_0^t a(t') dt', where t' is the variable of integration.
5. Evaluate the velocity at t = 1.0 \times 10^{-6} seconds (1.0 \mu s) using the expression for v(t) obtained from the integration.

Key Concepts

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. The strength of the electric field (E) is measured in newtons per coulomb (N/C) and can vary with time, as indicated in the question. In this case, the electric field is described by a linear function of time, which affects the force experienced by the proton.

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. For a proton, which has a positive charge, the force will act in the direction of the electric field. This force causes the proton to accelerate, which is crucial for determining its speed after a given time.

Kinematics and Acceleration

Kinematics is the branch of physics that describes the motion of objects. When a force acts on a particle, it causes acceleration, which can be calculated using Newton's second law (F = ma). The acceleration of the proton can be derived from the force acting on it due to the electric field, and this acceleration can be used to find the final speed after a specific time interval using kinematic equations.