Problem 81 - College Physics

Question

\(\bullet\) A singly charged ion of \(^{7} \mathrm{Li}\) (an isotope of lithium containing three protons and four neutrons) has a mass of \(1.16 \times\) \(10^{-26} \mathrm{kg} .\) It is accelerated through a potential difference of 220 \(\mathrm{V}\) and then enters a 0.723 T magnetic field perpendicular to the ion's path. What is the radius of the path of this ion in the magnetic field?

Step-by-Step Solution

Answer

The radius of the ion's path is approximately 0.018 meters.

Step 1: Determine the Charge of the Ion

As the ion is singly charged, it has a charge equal to the elementary charge, which is approximately \( q = 1.6 \times 10^{-19} \mathrm{C} \).

Step 2: Calculate the Ion's Velocity

The ion is accelerated through a potential difference \( V = 220 \mathrm{V} \). The kinetic energy gained by the ion is equal to the electric potential energy: \[ qV = \frac{1}{2}mv^2 \]Solve for \( v \):\[ v = \sqrt{\frac{2qV}{m}} = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 220}{1.16 \times 10^{-26}}} \]

Step 3: Calculate the Radius of the Ion's Path

The ion moves in a circular path within the magnetic field due to the Lorentz force acting as the centripetal force:\[ qvB = \frac{mv^2}{r} \]Rearrange and solve for \( r \):\[ r = \frac{mv}{qB} \]Use the calculated velocity from Step 2 and substitute all known values:\[ r = \frac{1.16 \times 10^{-26} \times v}{1.6 \times 10^{-19} \times 0.723} \]

Step 4: Calculate Final Value

Substitute the velocity \( v \) from Step 2 into the radius equation from Step 3 and calculate the numerical value for \( r \).Plugging in numbers:\[ \begin{align*} v & = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 220}{1.16 \times 10^{-26}}} \ r & = \frac{1.16 \times 10^{-26} \times v}{1.6 \times 10^{-19} \times 0.723} \end{align*} \] Evaluate to find \( r \).

Key Concepts

Singly Charged Ion

Consider a singly charged ion of lithium, specifically the isotope \(^{7} \mathrm{Li}\), which contains three protons and four neutrons. Even though neutrons don't carry a charge, the charge of this ion makes it important in physics and chemistry. A singly charged ion means it has either lost or gained exactly one electron. In the case of lithium, this usually involves losing one electron, resulting in a positive charge equal to the elementary charge.

• Elementary charge is denoted by \( q \) and is approximately \( 1.6 \times 10^{-19} \mathrm{C} \).
• As a positively charged ion, this lithium ion is denoted as Li\(^+\).

The notion of a "singly charged" ion is pivotal in understanding how the ion behaves under electric and magnetic fields. This basic information helps predict how the ion will react when subjected to external forces.

Potential Difference

Potential difference, often referred to as voltage, is the difference in electric potential between two points. When discussing the movement of charged particles, such as ions, through an electric field, this concept becomes very important.

• For a \(^{7} \mathrm{Li}\) ion, like the one considered in this problem, an increment in potential difference by 220 V results in an increase in kinetic energy.
• As charged particles are accelerated through a potential difference, they gain energy which is then converted into kinetic energy.

The relationship between potential difference and kinetic energy can be given by the equation \( qV = \frac{1}{2}mv^2 \), aligning the potential energy with kinetic energy for the motion of charges. This equation is important for calculating the speed of an ion like \( v = \sqrt{\frac{2qV}{m}} \). Therefore, increased potential difference leads to faster moving ions in the same unit mass.

Circular Motion in Magnetic Field

When the ion enters a magnetic field at an angle perpendicularly, as in our exercise, it follows a curved path. This is an example of circular motion resulting from the Lorentz force, which maintains the ion's trajectory.

• The significant point is that because the force acts perpendicular to the velocity, it keeps changing the direction, not the speed of the ion.
• This results in the ion following a circular path with a radius \( r \).

The radius of the ion’s path in a magnetic field is given by the equation \( r = \frac{mv}{qB} \). Mass \( m \), velocity \( v \), charge \( q \), and magnetic field strength \( B \) all influence the circular motion. This demonstrates the power of the magnetic field in directing charged particles and keeping them on a fixed curved path, making it possible to calculate the radius once the velocity is known.

Lorentz Force

Lorentz force is a fundamental concept in electromagnetism, describing the total force acting on charged particles moving in electric and magnetic fields. It combines both electric and magnetic field effects. For our singly charged \(^{7} \mathrm{Li}\) ion moving in just a magnetic field, the Lorentz force simplifies to the magnetic part alone. This force acts as the centripetal force required for circular motion in a magnetic field.

• The relation is captured by \( F = qvB \), where \( F \) is the force acting perpendicular to the velocity of the ion, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.
• The Lorentz force does not speed up or slow down the ion, rather it changes its trajectory, thus resulting in circular or spiral motion.

Understanding Lorentz force helps us grasp why charged particles curve under magnetic fields, and how it is crucial in creating controlled environments for charged particle motion, such as in cyclotrons and mass spectrometers.