Problem 25 - University Physics with Modern Physics

Question

LA A proton is placed in a uniform electric field of \(2.75 \times 10^{3} \mathrm{N} / \mathrm{C}\) . Calculate: (a) the magnitude of the electric force felt by the proton; (b) the proton's acceleration; (c) the proton's speed after 1.00\(\mu\) in the field, assuming it starts from rest.

Step-by-Step Solution

Answer

Force: \(4.40 \times 10^{-16}\,\text{N}\), Acceleration: \(2.63 \times 10^{11}\,\text{m/s}^2\), Speed: \(263\,\text{m/s}\).

Step 1: Calculate the Electric Force

The force exerted on a charged particle in an electric field can be calculated using the formula \( F = qE \), where \( q \) is the charge of the proton \( (1.60 \times 10^{-19} \text{ C}) \) and \( E \) is the electric field \( (2.75 \times 10^3 \text{ N/C}) \).\[ F = (1.60 \times 10^{-19} \, \text{C}) \times (2.75 \times 10^3 \, \text{N/C}) \]Calculate this product to find the force felt by the proton.

Step 2: Calculate the Proton's Acceleration

To find the acceleration, use Newton's second law, \( F = ma \), where \( m \) is the mass of a proton \( (1.67 \times 10^{-27} \, \text{kg}) \). Rearrange the formula to solve for acceleration \( a \): \( a = \frac{F}{m} \).Substitute the value of the force from Step 1 and the mass of the proton:\[ a = \frac{4.40 \times 10^{-16} \, \text{N}}{1.67 \times 10^{-27} \, \text{kg}} \]Perform the division to find the acceleration of the proton.

Step 3: Calculate the Proton's Speed After 1.00 Microsecond

The speed can be calculated using the formula \( v = u + at \), where \( u \) is the initial velocity (0 m/s), \( a \) is the acceleration found in Step 2, and \( t \) is the time \( (1.00 \, \mu\text{s} = 1.00 \times 10^{-6} \, \text{s}) \).\[ v = 0 + a \times 1.00 \times 10^{-6} \, \text{s} \]Substitute the value of acceleration and calculate \( v \).

Step 4: Conclusion

The magnitude of the electric force is \( 4.40 \times 10^{-16} \, \text{N} \), the acceleration is \( 2.63 \times 10^{11} \, \text{m/s}^2 \), and the speed of the proton after \( 1.00 \, \mu\text{s} \) is \( 263 \, \text{m/s} \).

Key Concepts

Proton Acceleration

When we talk about proton acceleration in an electric field, we're diving into a fundamental concept of physics. The acceleration of a charged particle, like a proton, is directly influenced by the forces acting on it. In this scenario, the force is due to the electric field.

Acceleration can be determined using Newton's second law, which states that force equals mass times acceleration: \( F = ma \). By rearranging the equation, we can solve for acceleration \( a \) as follows:

• \( a = \frac{F}{m} \)

Where \( F \) is the force exerted on the proton, and \( m \) is the mass of the proton. This tells us how fast the proton's velocity changes as it moves under the influence of the electric field. Remember, the greater the force, or the smaller the mass, the higher the acceleration.

In our specific exercise, once we calculate the force using the proton's charge and the electric field strength, we can find out precisely how quickly the proton will accelerate.

Electric Field

An electric field represents how an electrical charge exerts force on other charges around it. It's like an invisible "force field" that spreads out from charge sources.

In a uniform electric field, like the one in our exercise, the electric field lines are parallel and evenly spaced, which means that the force experienced by a charge within it is constant. The formula \( F = qE \) is employed to find the force on a charge:

• \( F \) is the force on the charge.
• \( q \) represents the charge of the proton, which is positive.
• \( E \) is the electric field strength given in Newtons per Coulomb (\( \text{N/C} \)).

This interaction is why a proton, when placed in an electric field, experiences a force that pushes it in the direction of the field.

Understanding electric fields helps us predict and control how charges behave, which is fundamental not only in academic settings but also in real-world applications like designing electronic circuits and even in understanding natural phenomena.

Fundamental Physics Equations

Physics is full of equations that help us understand and predict the universe's behavior. In this problem, we focus on a few key equations that simplify these processes.

The main equation here, \( F = qE \), allows us to calculate the force on a charge due to an electric field. It brings together the concepts of force, charge, and field strength.

Newton's second law, \( F = ma \), is another cornerstone. It connects force, mass, and acceleration, highlighting the direct relationship between these quantities.

Finally, the kinematic equation \( v = u + at \) lets us determine the speed of an object after a specific period, given constant acceleration. This formula helps calculate how the proton's speed will evolve over time, starting from rest and under constant acceleration.

• \( v \) is the final velocity.
• \( u \) is the initial velocity.
• \( a \) is the acceleration.
• \( t \) is the time interval.

Understanding these equations gives students the ability to solve a wide array of problems, not just those involving protons but in various domains of physics.