Question

Consider a long horizontally oriented conducting wire with \(\lambda=4.81 \cdot 10^{-12} \mathrm{C} / \mathrm{m} .\) A proton \(\left(\mathrm{mass}=1.67 \cdot 10^{-27} \mathrm{~kg}\right)\) is placed \(0.620 \mathrm{~m}\) above the wire and released. What is the magnitude of the initial acceleration of the proton?

Short answer

Answer: The magnitude of the initial acceleration of the proton is approximately \(1.32 \times 10^{15} \text{ m/s}^{2}\).

Step-by-step solution

Determine the electric field due to the conducting wire

We first need to find the electric field created by the wire at the proton's location. The magnitude of the electric field created by an infinitely long wire with a charge per unit length \(\lambda\) is given as: $$ E = \frac{2\lambda K}{r} $$ where \(K = 8.99 \times 10^9 \text{ N m}^2 / \text{C}^2\) is Coulomb's constant, and \(r\) is the distance from the proton to the wire.

Calculate the electric field

Now, we can plug in the given values to calculate the electric field at the proton's location: $$ E = \frac{2(4.81 \times 10^{-12}\text{ C/m}) (8.99 \times 10^{9}\text{ N m}^2 / \text{C}^2)}{0.620\text{ m}} $$ Which gives: $$ E = 1.38\times 10^7 \text{ N/C} $$

Determine the electric force on the proton

The electric force on the proton can be calculated by using: $$ F_{electric} = q_e \times E $$ where \(q_e =1.6 \times 10^{-19} \text{ C} \) is the charge of the proton. Plugging in the obtained value of the electric field, we get: $$ F_{electric} = (1.6 \times 10^{-19}\text{ C}) (1.38\times 10^7\text{ N/C}) $$ Which gives: $$ F_{electric} = 2.21\times 10^{-12} \text{ N} $$

Determine the weight of the proton

We can calculate the weight of the proton by multiplying its mass by the gravitational acceleration: $$ W = m_{proton} \times g $$ where \(m_{proton} = 1.67 \times 10^{-27} \text{ kg}\) and \(g = 9.81 \text{ m/s}^{2}\). Substituting the known values, we get: $$ W = (1.67 \times 10^{-27}\text{ kg})(9.81 \text{ m/s}^{2}) $$ Which gives: $$ W = 1.64 \times 10^{-26} \text{ N} $$

Apply Newton's second law to find the acceleration

Now, we can apply Newton's second law to calculate the vertical acceleration of the proton: $$ \sum F = m_{proton} \times a $$ The total vertical force acting on the proton (\(\sum F\)) is the difference between the electric force and its weight, since the electric force is attractive and the weight is downward: $$ \sum F = F_{electric} - W $$ Plugging the known values, we get: $$ (2.21\times 10^{-12} \text{ N}) - (1.64 \times 10^{-26} \text{ N})= m_{proton} \times a $$ Now solving for the acceleration \(a\): $$ a = \frac{(2.21\times 10^{-12} \text{ N}) - (1.64 \times 10^{-26} \text{ N})}{1.67 \times 10^{-27}\text{ kg}} $$ Which gives: $$ a = 1.32 \times 10^{15} \text{ m/s}^{2} $$ The magnitude of the initial acceleration of the proton is approximately \(1.32 \times 10^{15} \text{ m/s}^{2}\).

Key concepts

Understanding the Electric Field

When we talk about electric fields, we're referring to a kind of 'invisible force field' that surrounds electric charges. It's a vector field that relates to the electric force that a charge would experience at any point in space. The strength of the field is determined by the amount of charge and the distance from the charge.

For instance, the problem presented describes a long conducting wire carrying a linear charge density \( \lambda \). This wire creates an electric field around it. By using the given formula \( E = \frac{2\lambda K}{r} \) where \( r \) is the distance from the wire to the charge, we can calculate the magnitude of this field. \( K \) represents Coulomb's constant, which tells us about the strength of the electric interaction.

Calculating the electric field correctly is critical because it directly informs us about the magnitude of the force that will act on the proton. It's important to understand that the electric field is a representation of the influence that a charge distribution exerts on the space around it, affecting other charges present within that field.

Coulomb's Law and its Role in Physics Problems

Coulomb's law is essential in understanding electric interactions between charged particles. It describes the amount of force between two point charges at a distance. The law states that the force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them. The constant \( K \) often found in such equations is a proportionality constant known as Coulomb's constant, carrying the value \( 8.99 \times 10^9 \text{ N m}^2 / \text{C}^2 \) in vacuum.

In the context of the mentioned physics problem, Coulomb's law is indirectly applied through the electric field equation for a continuous charge distribution, rather than a point charge. Understanding how this law scales from individual charges to a continuous distribution like the wire is central to solving such problems. It's this foundational law that allows us to calculate forces in numerous electrostatic situations, including that of a proton near a charged wire.

Applying Newton's Second Law

Newton's second law of motion describes how the velocity of an object changes when it is subjected to an external force. The law is expressed by the equation \( F = ma \), where \( F \) is the total force applied, \( m \) is the mass of the object, and \( a \) is the acceleration. This is a cornerstone in physics as it connects force, mass, and acceleration in a straightforward and quantifiable way.

In our proton's scenario, we use Newton's second law to figure out how much the proton accelerates due to the forces acting on it. The proton's acceleration is the result of the net force applied to it which, in this case, is the electric force upward and gravity force downward. By subtracting the weight of the proton (due to gravity) from the electric force and dividing by the mass, we get the acceleration. The ability to predict the behavior of an object under various forces is an invaluable tool in the realm of physics and engineering.

Proton Acceleration in an Electric Field

When it comes to proton acceleration in an electric field, we must consider two main forces: the electrical force pulling the proton toward the wire and the proton's weight, which is the force exerted by gravity pulling it down. The actual acceleration of the proton occurs due to the net force, which is the vector sum of all the forces acting on it.

An important takeaway is that acceleration is the rate of change of velocity. When a proton is released from rest, as described in the problem, the initial acceleration is entirely due to the electric force, since its initial velocity is zero. Here, we apply Newton's second law to identify the relationship between the net force and acceleration. The concept of a proton's acceleration is fundamental to fields like particle physics and is also used in devices like accelerators, which have implications ranging from fundamental research to medical treatments such as cancer therapy.