Problem 121 - A Graphical Approach to College Algebra

Question

Particle 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 \(-4 \times 10^{-17}\) meters.

Step 1: Identify the Given Information

We are given the equation of the path of an alpha particle between charged plates: \(y = -\frac{k}{2 v_0} x^2\). The constants and variables given are:- \(k = 5 \times 10^{-9}\)- \(v_0 = 10^7\) meters per second- The distance \(x = 0.4\) meters.

Step 2: Substitute Known Values into the Equation

Insert the values of \(k\), \(v_0\), and \(x\) into the equation to find the value of \(y\):\[ y = -\frac{5 \times 10^{-9}}{2 \times 10^7} (0.4)^2 \]

Step 3: Calculate the Expression

Compute the value of the expression:1. Calculate \(2 \times 10^7 = 2 \times 10^7 = 2 \times 10000000 = 20000000\).2. Calculate \(x^2 = (0.4)^2 = 0.16\).3. Substitute these values back into the expression: - \( y = -\frac{5 \times 10^{-9} \times 0.16}{20000000}\)

Step 4: Simplify the Expression

Determine the result by simplifying the fraction:\[ y = -\frac{8 \times 10^{-10}}{2 \times 10^7} = -\frac{8 \times 10^{-10}}{20000000} \]Multiply and simplify:\[ y = -\frac{8}{2} \times 10^{-10} \times 10^{-7} \]\[ y = -4 \times 10^{-17} \]

Step 5: Final Answer Interpretation

The deflection of the alpha particle in the \(y\)-direction is calculated to be \(-4 \times 10^{-17}\) meters. The negative sign indicates the direction of deflection.

Key Concepts

Alpha Particle

The alpha particle is a type of subatomic particle that consists of two protons and two neutrons, making it essentially a helium nucleus. These particles are emitted naturally during the radioactive decay of some elements and can also be produced artificially in nuclear reactions.

• They are positively charged due to the protons they contain.
• They are relatively heavy among particles in the nuclear realm.
• They possess high kinetic energy but do not penetrate materials deeply due to their larger mass and charge.

When alpha particles travel, they can be influenced by electric and magnetic fields. This property makes them useful in experimental physics, where their movement can be tracked and analyzed to understand various forces or fields they pass through.

Equation of Path

Considering the motion of an alpha particle between charged plates leads to a parabolic trajectory, which can be described mathematically. The path is influenced by the electric field between the plates, and the motion in the vertical direction can be represented by parabolic equations.
In our context, the equation used is:\[ y = -\frac{k}{2 v_{0}} x^{2}\]
The variables in this equation represent:

• \(y\): the vertical deflection of the particle.
• \(x\): the horizontal distance traveled by the alpha particle.
• \(k\): a constant related to the strength of the electric field.
• \(v_0\): the initial velocity of the particle.

The negative sign in the equation indicates the direction of deflection relative to the initial trajectory, important for understanding in which way the particle bends.

Deflection Calculation

Deflection refers to how much the alpha particle's path deviates from its original course due to the influence of external forces, like the electric field in this case.
Calculating this deflection involves using the given equation of path and inserting known values:

• Substitute into the equation: \(y = -\frac{5 \times 10^{-9}}{2 \times 10^{7}} (0.4)^2\)
• Simplify and solve: The computation results in \(y = -4 \times 10^{-17}\) meters.

This result tells us two things:

• The exact measure of how much the particle's path is perturbed by the charged plates.
• The negative sign shows the deflection direction, which helps indicate the electric field's influence.

Charged Plates

Charged plates create an electric field, which is instrumental in deflecting the alpha particle's trajectory. These plates are commonly used in experiments to study particle behavior under electric forces.

• The polarity of the plates determines the direction in which particles are deflected.
• The strength of the electric field will affect how much the path bends.
• A 2000-volt potential difference between the plates indicates significant energy, influencing particle motion considerably.

In practical applications, such an arrangement helps physicists understand foundational principles of electromagnetism and particle physics by observing the interactions of charged particles with fields.