Problem 34 - University Physics with Modern Physics

Question

An infinitely long line of charge has linear charge density \(5.00 \times 10^{-12} \mathrm{C} / \mathrm{m} .\) A proton (mass \(1.67 \times 10^{-27} \mathrm{kg}\) . charge \(+1.60 \times 10^{-19} \mathrm{C} )\) is 18.0 \(\mathrm{cm}\) from the line and moving directly toward the line at \(1.50 \times 10^{3} \mathrm{m} / \mathrm{s}\) (a) Calculate the proton's initial kinetic energy. (b) How close does the proton get to the line of charge? (Hints See Example \(23.10 . )\)

Step-by-Step Solution

Answer

(a) Initial kinetic energy: \(1.88 \times 10^{-21} \, \mathrm{J}\). (b) Proton gets as close as \(0.051\,\mathrm{m}\) to the line.

Step 1: Calculate the Initial Kinetic Energy

To find the initial kinetic energy of the proton, we use the formula for kinetic energy: \[ KE = \frac{1}{2} mv^2 \]where:- \(m = 1.67 \times 10^{-27}\,\mathrm{kg}\) (mass of the proton)- \(v = 1.50 \times 10^{3}\,\mathrm{m/s}\) (initial velocity of the proton)Substitute these values into the formula:\[ KE = \frac{1}{2} \times 1.67 \times 10^{-27} \times (1.50 \times 10^{3})^2 \]Calculating this gives:\[ KE \approx 1.88 \times 10^{-21}\,\mathrm{J} \]

Step 2: Calculate the Electric Field due to the Line of Charge

The electric field \(E\) from an infinite line of charge with linear charge density \(\lambda\) is given by:\[ E = \frac{\lambda}{2\pi \varepsilon_0 r} \]where:- \(\lambda = 5.00 \times 10^{-12}\,\mathrm{C/m}\)- \(r = 0.18\,\mathrm{m}\) (initial distance from the line)- \(\varepsilon_0 = 8.85 \times 10^{-12}\,\mathrm{C^2/(N \cdot m^2)}\) (vacuum permittivity)Substitute the values:\[ E = \frac{5.00 \times 10^{-12}}{2\pi \times 8.85 \times 10^{-12} \times 0.18} \]Calculating this gives:\[ E \approx 4.97 \times 10^{4}\,\mathrm{N/C} \]

Step 3: Compute the Potential Energy at Initial Position

The potential energy \(U\) of a point charge in an electric field is given by:\[ U = q \cdot V \]where \(V = -\int E\,dr = E \cdot r\) for an infinite line of charge (since the line is infinitely long and uniformly charged, we can use the above relation derived from one integral over the field). Therefore:\[ U = q \cdot E \cdot r \]with:- \(q = 1.60 \times 10^{-19}\,\mathrm{C}\) (charge of the proton)Substitute the values at initial distance:\[ U = 1.60 \times 10^{-19} \times 4.97 \times 10^{4} \times 0.18 \]Calculating this gives:\[ U \approx 1.43 \times 10^{-18}\,\mathrm{J} \]

Step 4: Determine the Minimum Distance from the Line of Charge

At the closest point to the line, the kinetic energy of the proton will be zero as all kinetic energy would convert to potential energy. We need to use energy conservation:Initial Kinetic Energy + Initial Potential Energy = Potential Energy at the closest approachLet \(r_{min}\) be the closest distance to the line, then:\[ 1.88 \times 10^{-21} + 1.43 \times 10^{-18} = q \cdot E \cdot r_{min} \]Solving for \(r_{min}\):\[ r_{min} = \frac{1.88 \times 10^{-21} + 1.43 \times 10^{-18}}{1.60 \times 10^{-19} \times 4.97 \times 10^{4}} \]Calculating this gives:\[ r_{min} \approx 0.051\,\mathrm{m} \]

Key Concepts

Electric Field

An electric field is a concept invented to express how a charged object exerts force on another charged object. To better understand this, imagine a space around any charged object filled with invisible lines representing the electric field. When another charge is placed near this field, it "feels" a force due to the electric field. The strength of the electric field depends on several factors. One important factor is the source charge's amount and distribution. In the exercise example, we have an infinitely long line of charge. This creates a uniform electric field around it, which we can calculate using the equation:

• \( E = \frac{\lambda}{2\pi \varepsilon_0 r} \)

Here, \(\lambda\) is the charge per unit length or linear charge density, \(\varepsilon_0\) is the permittivity of free space, and \(r\) is the distance from the line of charge. The electric field strength points radially outward from the line, and it decreases with distance, meaning closer objects will feel a stronger force. This field plays a key role in influencing the motion and energy of charges like protons, affecting their potential and kinetic energy as discussed in further sections.

Potential Energy

Potential energy in electrostatics is the energy stored due to the position of a charged particle in an electric field. Imagine it as the energy you have by virtue of your position on a hill—a charged particle has potential energy when placed within an electric field. This potential energy stems from the work done against the electric field when moving a charge. In our exercise, this concept helps us understand how the proton's energy changes as it moves closer to or away from the line of charge. We calculate it as:

• \( U = q \cdot E \cdot r \)

Here, \(q\) is the charge of the particle, \(E\) is the electric field strength, and \(r\) is the distance from the line. This shows that the potential energy is directly proportional to both the field strength and the distance from the charge.
The interesting part about potential energy is how it converts into other forms like kinetic energy. As the proton moves through the field, its potential energy decreases as it gets closer to the line of charge, converting into kinetic energy until it reaches its closest point where only potential energy remains. Understanding this transformation is crucial for describing the proton's motion in the electric field.

Kinetic Energy

Kinetic energy describes the energy of motion. Imagine how a rolling ball picks up speed as it moves down a hill. In electrostatic terms, a charged particle like a proton accelerates as it moves closer to a source charge, converting its stored potential energy into kinetic energy. This conversion happens due to the attractive or repulsive forces from the electric field.
The formula used to find kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

where \(m\) is the mass of the proton and \(v\) its velocity. We calculate the initial kinetic energy of the proton based on its initial speed. As the proton moves toward the charged line, its kinetic energy changes depending on how much of its potential energy converts to kinetic energy. Close to the electric field, as potential energy increases and reaches a maximum point, the kinetic energy will diminish to zero momentarily before it reverses direction. This energy balance is what dictates how far the proton travels and how fast it moves in an electric field, showing the fascinating dance between these two forms of energy.