CALC A uniform electric field’s strength is increasing with time as $E=(1.5\times {10}^4+(5.0\times {10}^{10}{\text{s}}^{-1}\left)t\right)\text{N/C}$. 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. Step 1: Understand the problem. The electric field is time-dependent, given by \( E = 1.5 \times 10^4 + (5.0 \times 10^{10} \text{ s}^{-1})t \) N/C. A proton is released from rest at \( t = 0 \), and we need to calculate its speed after \( t = 1.0 \mu \text{s} \) (\( 1.0 \times 10^{-6} \text{s} \)).
2. Step 2: Recall the relationship between force and electric field. The force on a proton due to the electric field is \( F = qE \), where \( q \) is the charge of the proton (\( q = 1.6 \times 10^{-19} \text{ C} \)). Substitute the time-dependent electric field \( E(t) \) into this equation to express \( F(t) \).
3. Step 3: Use Newton's second law \( F = ma \) to find the acceleration of the proton. The mass of the proton is \( m = 1.67 \times 10^{-27} \text{ kg} \). Combine \( F(t) = qE(t) \) and \( F = ma \) to express the acceleration as \( a(t) = \frac{qE(t)}{m} \).
4. Step 4: Integrate the acceleration \( a(t) \) over time to find the velocity \( v(t) \). Since the proton starts from rest, the initial velocity \( v(0) = 0 \). The velocity at time \( t \) is given by \( v(t) = \int_0^t a(t') dt' \). Substitute \( a(t') = \frac{q}{m} \left( 1.5 \times 10^4 + (5.0 \times 10^{10} \text{ s}^{-1})t' \right) \) into the integral.
5. Step 5: Evaluate the integral to find \( v(t) \) at \( t = 1.0 \mu \text{s} \). Perform the integration step-by-step, keeping the constants \( q \) and \( m \) outside the integral. After integration, substitute \( t = 1.0 \times 10^{-6} \text{s} \) into the resulting expression for \( v(t) \).

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 deals with the motion of objects. When a charged particle like a proton is subjected to a force, it accelerates according to Newton's second law (F = ma). The acceleration can be calculated from the force acting on the proton, and knowing the initial conditions (like starting from rest) allows us to use kinematic equations to find the final speed after a specified time.