Question

An electron with a speed of $4.0\cdot {10}^5\mathrm{m}/\mathrm{s}$ enters a uniform magnetic field of magnitude $0.040\mathrm{T}$ at an angle of ${35}^{\circ }$ to the magnetic field lines. The electron will follow a helical path. a) Determine the radius of the helical path. b) How far forward will the electron have moved after completing one circle?

Short answer

The electron will have moved forward 1.16 * 10^-4 meters after completing one circle.

Step-by-step solution

Calculate the perpendicular component of the velocity

To find the perpendicular component of velocity, we'll use trigonometry and the given angle between the velocity and magnetic field lines (35°): v⊥ = v * sin(35°) Substitute the given value of electron speed: v⊥ = (4.0 * 10^5 m/s) * sin(35°) v⊥ ≈ 2.3 * 10^5 m/s

Calculate the radius of the helical path

Using the formula for helical path radius, we have: r = mv⊥/qB The mass of an electron (m) = 9.11 * 10^-31 kg The charge of an electron (q) = 1.6 * 10^-19 C Now plug in the values of v⊥, m, q, and B into the formula: r ≈ (9.11 * 10^-31 kg * 2.3 * 10^5 m/s) / (1.6 * 10^-19 C * 0.040 T) r ≈ 3.3 * 10^-4 m So, the radius of the helical path is 3.3 * 10^-4 meters.

Calculate the parallel component of the velocity

Similarly, we can find the parallel component of velocity by using the cosine of the given angle between the velocity and magnetic field lines (35°): v∥ = v * cos(35°) Substituting the given value of electron speed: v∥ = (4.0 * 10^5 m/s) * cos(35°) v∥ ≈ 3.3 * 10^5 m/s

Calculate the period of the circular motion

T = 2 * π * m / (q * B) Now plug in the values of m, q, and B into the formula: T ≈ 2 * π * (9.11 * 10^-31 kg) / (1.6 * 10^-19 C * 0.040 T) T ≈ 3.52 * 10^-10 s

Calculate the distance moved forward after completing one circle

Use the formula for the pitch of the helix: p = v∥T Now plug in the values of v∥ and T into the formula: p ≈ (3.3 * 10^5 m/s) * (3.52 * 10^-10 s) p ≈ 1.16 * 10^-4 m So, the electron will have moved forward 1.16 * 10^-4 meters after completing one circle. In conclusion, the radius of the helical path is 3.3 * 10^-4 meters, and the electron will have moved forward 1.16 * 10^-4 meters after completing one circle.

Key concepts

Magnetic Fields

Magnetic fields are invisible forces that exert a magnetic influence on certain materials and charged particles like electrons.
They are typically depicted by magnetic field lines, which show the direction and relative strength of the field. When a charged particle enters a magnetic field, the field exerts a force on it. This force is perpendicular to both the velocity of the particle and the direction of the magnetic field itself. This phenomenon is a result of the Lorentz force, described by the formula:$$F=q(v\times B)$$where:- $F$ is the magnetic force,- $q$ is the charge of the particle,- $v$ is the velocity vector,- $B$ is the magnetic field vector.In our exercise, an electron, which is a negatively charged particle, enters a magnetic field at an angle to the field lines. As a result of the perpendicular force, the electron undergoes a specific kind of motion known as helical motion.

Electron Dynamics

Electron dynamics describe how electrons move under various forces. When these subatomic particles enter a magnetic field, their motion is altered.
The key characteristics of this motion are determined by the initial speed and direction of the electron with respect to the magnetic field lines. For instance, the speed of the electron in our example is given as $4.0\times {10}^5\text{m/s}$, and the angle with the magnetic field is ${35}^{\circ }$. This setup allows us to split the electron's velocity into two components:- **Perpendicular Component ($v_{\perp }$)**: This is calculated using the sine of the angle. It represents the component of the electron's velocity that is affected by the magnetic field force, causing circular motion.- **Parallel Component ($v_{\parallel }$)**: Determined using the cosine of the angle, this component remains constant as it moves along the direction of the magnetic field. Understanding these components helps visualize and calculate the path of the electron as it spirals along its helical path.

Circular Motion

Circular motion involves an object moving in a circular path due to a centripetal force.
In the case of electrons in a magnetic field, the perpendicular component of their velocity causes them to move in a circle.The formula for radius of this circular path is:$$r=\frac{m{v}_{\perp }}{qB}$$where:- $m$ is the mass of the electron,- $v_{\perp }$ is the perpendicular velocity component,- $q$ is the charge,- $B$ is the magnetic field strength.For electrons, their small mass and negative charge mean they can have a relatively tight circular path when in a magnetic field. In the given example, once calculated, the radius is computed as $3.3\times {10}^{-4}\text{m}$. This demonstrates how the tiny charge and mass of an electron allow it to move in a tight circle under the influence of a relatively modest magnetic field.

Physics Problem Solving

Physics problem solving is all about breaking down complex problems into manageable steps.
For the problem of helical motion of electrons, the steps include understanding the forces at play and using known formulas to solve for unknowns. Here are the essential steps to solve similar problems:

• **Identify given quantities**: such as speed, magnetic field strength, angles, and physical constants like mass and charge of the electron.
• **Determine components of velocity**: Use trigonometric functions to resolve the initial speed into perpendicular and parallel components relative to the magnetic field.
• **Apply formulas systematically**: For radius, use the helical path radius formula; for displacement after a circle, compute using pitch of the helix.
• **Review each step carefully**: After calculation, check results for accuracy and consistency with physical reality.

Utilizing this structured approach enhances your comprehension and ensures effective problem-solving techniques in physics.