Problem 118 - A Graphical Approach to Precalculus with Limits

Question

When an alpha particle (a subatomic particle) is moving in a horizontal path along the positive \(x\) -axis and passes between charged plates, it is deflected in a parabolic path. If the plate is charged with 2000 volts and is 0.4 meter long, then an alpha particle's path can be described by the equation $$y=-\frac{k}{2 v_{0}} x^{2}$$ where \(k=5 \times 10^{-9}\) is constant and \(v_{0}\) is the initial velocity of the particle. If \(v_{0}=10^{7}\) meters per second, what is the deflection of the alpha particle's path in the y-direction when \(x=0.4\) meter? (Source: Semat, H. and J. Albright, Introduction to Atomic and Nuclear Physics, Holt, Rinehart and Winston.)

Step-by-Step Solution

Answer

The deflection is \( y = -4 \times 10^{-10} \) meters.

Step 1: Substitute Values into the Equation

The equation given is \( y = -\frac{k}{2 v_{0}} x^{2} \). Substitute \( k = 5 \times 10^{-9} \), \( v_{0} = 10^{7} \), and \( x = 0.4 \) into the equation.

Step 2: Calculate the Constant Term

At this point, simplify the fraction \(-\frac{k}{2 v_{0}}\) by substituting \( k = 5 \times 10^{-9} \) and \( v_{0} = 10^{7} \). Compute \(-\frac{5 \times 10^{-9}}{2 \times 10^{7}}\).

Step 3: Perform the Multiplication with \( x^2 \)

First, compute \( x^2 \), which is \( 0.4^2 = 0.16 \). Then, multiply the result of the previous step by \( 0.16 \).

Step 4: Calculate the Final Deflection

Multiply the computed constant term with \( 0.16 \) to find \( y \). This represents the deflection in the y-direction when \( x = 0.4 \) meter.

Key Concepts

Alpha Particle

An alpha particle is a type of subatomic particle that plays a crucial role in the study of nuclear physics. It consists of two protons and two neutrons, giving it a similar composition to a helium nucleus.
The charge of an alpha particle is positive, which results in its interaction with electric fields, causing deflection.

• These particles are released during radioactive decay of certain isotopes, such as uranium and radium.
• Despite their strong charge, alpha particles have a relatively large mass, which restricts their penetration power as compared to other particles like beta particles or gamma rays.

When alpha particles travel through a medium, their path can be altered by external forces. For instance, as they pass through charged plates, like in the exercise mentioned, they undergo deflection.
This deflected motion often adopts a parabolic path generally due to the influence of an electric field, which causes differences in velocity along the x and y directions.

Deflection Calculation

In the context of the exercise, calculating the deflection of an alpha particle while it travels between charged plates involves some straightforward physics equations.
The path in which the particle is deflected is modeled using the equation: \( y = -\frac{k}{2 v_{0}} x^{2} \).
To perform the calculation:

• First, identify the constants given, such as \(k = 5 \times 10^{-9}\) and initial velocity \(v_{0} = 10^{7}\) meters per second.
• Substitute these values along with the particular x-position you are interested in (\(x = 0.4\) meters) into the equation.
• Calculate the fraction \(-\frac{k}{2 v_{0}}\), simplifying it with \(k\) and \(v_{0}\)'s given values.
• Continue by computing \(x^2\), which, in this case, is \(0.16\), then multiply this with your previously obtained fraction to get the particle's deflection in the y-axis.

This process allows one to determine how much the particle's path is altered when it traverses the charged field.

Initial Velocity

Initial velocity, denoted as \(v_{0}\), is a fundamental concept in physics that defines the speed at which an object begins its motion.
In this particular exercise, the alpha particle's initial velocity is essential, as it affects the magnitude of the deflection.
The relationship between initial velocity and deflection can be understood through these points:

• The initial velocity \(v_{0} = 10^{7}\) m/s indicates how fast the alpha particle is moving before it encounters the electric field.
• Higher initial velocities generally lead to lesser deflections, as the kinetic energy of the particle allows it to maintain a more straightforward path amid external forces.
• In contrast, lower initial velocities would cause the particle to follow the influence of the electric field more pronouncedly, leading to greater deflection.

In conclusion, understanding initial velocity is vital for predicting the behavior of particles when subjected to deflecting forces, such as electric fields.