91Ó°ÊÓ

It is not possible to see very small objects, such as viruses, using an ordinary light microscope. An electron microscope can view such objects using an electron beam instead of a light beam. Electron microscopy has proved invaluable for investigations of viruses, cell membranes and subcellular structures, bacterial surfaces, visual receptors, chloroplasts, and the contractile properties of muscles. The "lenses" of an electron microscope consist of electric and magnetic fields that control the electron beam. As an example of the manipulation of an electron beam, consider an electron traveling away from the origin along the \(x\) axis in the \(x y\) plane with initial velocity \(\mathbf{v}_{i}=v_{i}\) i. As it passes through the region \(x=0\) to \(x=d\), the electron experiences acceleration \(\mathbf{a}=a_{x} \mathbf{i}+a_{y} \mathbf{j},\) where \(a_{x}\) and \(a_{y}\) are constants. For the case \(v_{i}=1.80 \times 10^{7} \mathrm{m} / \mathrm{s}\) \(a_{x}=8.00 \times 10^{14} \mathrm{m} / \mathrm{s}^{2}\) and \(a_{y}=1.60 \times 10^{15} \mathrm{m} / \mathrm{s}^{2},\) determine at \(x=d=0.0100 \mathrm{m}\) (a) the position of the electron, (b) the velocity of the electron, (c) the speed of the electron, and (d) the direction of travel of the electron (i.e., the angle between its velocity and the \(x\) axis).

Step by step solution

01

Setup the 2D motion equations

We will handle the \(x\)-direction and \(y\)-direction independently. The position in the \(x\)-direction, \(x\), is given by \(x = x_i + v_{ix}t + 0.5 a_{x}t^2\). The position in the \(y\)-direction, \(y\), is given by \(y = y_i + v_{iy}t + 0.5 a_{y}t^2\). Here, \(x_i = 0, y_i = 0\) (as the electron starts from the origin), \(v_{ix} = 1.80 * 10^7 m/s, v_{iy} = 0\) (as the electron begins moving solely along the \(x\)-axis), \(a_{x} = 8.00 * 10^{14} m/s^2, a_{y} = 1.60 * 10^{15} m/s^2\). We need to calculate the time \(t\) the electron travels a distance \(x = d = 0.0100 m\).

02

Calculate the time it takes the electron to travel the distance

Substitute into the \(x\)-direction position equation gives \(d = v_{ix}t + 0.5 a_{x}t^2\). Solving for \(t\) in terms of given quantities gives \(t = sqrt((2d)/a_{x})\). Substitute into this equation the given data \(d = 0.0100 m\) and \(a_{x} = 8.00 * 10^{14} m/s^2\) gives \(t = 1.27 * 10^{-8} s\).

03

Calculate the position of the electron

Substitute \(t = 1.27 * 10^{-8} s\) into the \(y\)-direction position equation gives \[y = 0.5a_{y}t^2\]. Substituting \(a_y = 1.60 * 10^{15} m/s^2\) gives \(y = 1.29 * 10^{-2} m\). Therefore, the position of the electron is \((0.0100, 1.29 * 10^{-2}) m\).

04

Calculate the velocity of the electron

The \(x\) and \(y\) components of velocity can be obtained from their respective velocity equations: \(v_x = v_{ix} + a_{x}t\) and \(v_y = v_{iy} + a_{y}t\). Substituting the known values gives \(v_x = -2.42 * 10^{7} m/s\) and \(v_y = 2.03 * 10^{7} m/s\). Therefore, the velocity of the electron is \((-2.42 * 10^{7}, 2.03 * 10^{7}) m/s\).

05

Calculate the speed of the electron

The speed of the electron is given by the magnitude of the velocity vector: \(v = sqrt(v_x^2 + v_y^2)\). Substituting the known values gives \(v = 3.18 * 10^{7} m/s\).

06

Calculate the direction of travel of the electron

The direction of the velocity vector can be calculated by the equation \(tan\theta = v_y/v_x\). Solving for \(\theta\) gives \(\theta = atan(v_y/v_x) = 40.3^\circ\). However, since \(v_x\) is negative and \(v_y\) is positive, the velocity vector falls into the second quadrant, so the final direction of the electron \(= 180^\circ - 40.3^\circ = 139.7^\circ\) ccw from the positive \(x\)-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Beam

An electron beam is central to the function of an electron microscope, allowing it to visualize objects that are too small for light microscopes to detect. In an electron microscope, the beam consists of electrons—subatomic particles with negative charge—which can be focused and manipulated using magnetic and electric fields. This focused beam is directed at a sample, and its interactions with the sample contribute to forming a highly magnified image.

Electron beams are incredibly precise, permitting the examination of tiny structures such as viruses, cell membranes, and subcellular components that are invisible to traditional optical technologies. This is because electrons have much shorter wavelengths compared to visible light, providing higher resolution. The movement and direction of the electron beam are fundamental to achieving proper magnification and image clarity.

Electric and Magnetic Fields

Electric and magnetic fields are the "lenses" of the electron microscope, crucial for steering and focusing the electron beam. These fields control the electrons' paths by exerting forces on them, based on their charge.

**Electric Fields:** These fields work by exerting force on the negatively charged electrons, causing them to accelerate in a specific direction. The manipulation through electric fields determines the speed and kinetic energy of the electron beam.

**Magnetic Fields:** Magnetic fields influence electrons differently, as they affect the direction by which the electrons travel. When electrons move through a magnetic field, they experience a force perpendicular to their velocity, causing them to curve. This curving effect is pivotal for focusing the beam onto the sample and ensures that the image produced is focused and accurate.

Managing these fields accurately allows for controlled and precise imaging in electron microscopy, achieving the high resolution necessary for viewing microscopic structures.

2D Motion Equations

In this context, 2D motion equations are used to describe and predict the path and behavior of the electron as it moves through the microscope's fields. As the electrons travel in a plane, their motion can be split into two independent perpendicular directions: the x-direction and the y-direction. This helps in analyzing their movement.

**Position Equations:**
- The equation for the position in the x-direction is: \( x = x_i + v_{ix}t + 0.5 a_{x}t^2 \).
- For the y-direction position: \( y = y_i + v_{iy}t + 0.5 a_{y}t^2 \).
Here, the initial conditions, velocities, and accelerations are plugged into these equations to find the position at any given time.

These equations are derived from the basic kinematic equations of motion, assuming constant acceleration. By solving these, we can determine not only where the electron is at a certain time but also how it moves through the electric and magnetic fields present in an electron microscope.

Electron Velocity

The velocity of an electron in an electron microscope is a crucial component determining how the electron beam interacts with the sample. Velocity has both a magnitude (speed) and direction, and when an electron passes through fields with acceleration, its velocity changes over time.

**Velocity Components:**
- In the x-direction, the velocity can be calculated with \( v_x = v_{ix} + a_{x}t \).
- In the y-direction, it's given by \( v_y = v_{iy} + a_{y}t \).
These equations incorporate initial velocities and the influence of acceleration in each direction.

Once the velocity components are calculated, the speed of the electron, which is the magnitude of the velocity vector, can be calculated using \( v = \sqrt{v_x^2 + v_y^2} \). The direction of the electron's velocity vector is sought using trigonometric functions, specifically by calculating the angle with respect to the initial direction of motion.

Understanding the dynamics of electron velocity allows scientists to fine-tune the imaging process and improve resolution, making electron microscopy an invaluable resource for visualizing minute samples.