Physics Study Guide for the HESI Exam

Optics

Optics is the study of how light behaves when it reflects, refracts, or passes through lenses and mirrors. Understanding how light moves through mirrors and lenses is vital for various scientific endeavors, including astronomy and optometry. There are two main types of mirrors and lenses that you will need to be familiar with: concave and convex.

Concave Mirrors and Lenses

Concave mirrors and lenses are curved inward, like the inside of a bowl, and they focus light to a point.

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For an object placed in front of a concave mirror, the relationship between the image distance \((d_i)\), the object distance \((d_o)\) and the focal length (\(f\)) is expressed as follows:

\[\frac{1}{f} = \frac{1}{d_o} +\frac{1}{d_i}\]

The focal length ends at the focal point (\(F\)), which is where the light converges. By convention, the signs are negative if the distances are measured to the left (front) of the mirror.

The magnification (\(M\)) of that image is defined as:

\[M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\]

where \(h_i\) is the image height and \(h_o\) is the object height.

Here is an example problem.

Find the image height of a \(5\text{ cm}\) high object, placed \(10\text{ cm}\) away from a concave mirror with a focal length of \(20\text{ cm}\).

Solution

We know \(d_i = 10\text{ cm}\) and \(f = 20\text{ cm}\), so the first equation can be used to find \(d_o\):

\[\frac{1}{20} = \frac{1}{d_o} +\frac{1}{10}\] \[\frac{1}{d_o} = \frac{1}{20} - \frac{1}{10}\] \[\frac{1}{d_o} = -\frac{1}{20}\] \[d_o = -20 \text{ cm}\]

We also know \(h_o = 5\text{ cm}\), so the second equation can now be used to find the image height:

\[\frac{h_i}{5} = -\frac{10}{-20}\] \[h_i = 5 \cdot 0.5 = 2.5 \text{ cm}\]

Convex Mirrors and Lenses

Convex mirrors curve outward and spread light rays apart. The images they form are always virtual, upright, and smaller.

Retrieved from:https://openstax.org/books/physics/pages/16-3-lenses

Convex mirrors use the same equations as concave mirrors, but the focal length is negative. In concave mirrors, the images formed depend on the position of the object. They can be real or virtual, upright or inverted, and reduced or enlarged in size.

We’ll do an example with a convex mirror.

A \(4\text{ cm}\) object is placed \(12\text{ cm}\) from a convex mirror with a focal length of \(–18\text{ cm}\). Find the image height.

Solution

As before, we’ll use the first formula from above to determine the object distance:

\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\] \[\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{-18} - \frac{1}{12} \approx -0.1389\] \[d_i = \frac{1}{-0.1389} = -7.2 \text{ cm}\]

Now, we can find the image height with the second formula:

\[M = -\frac{d_i}{d_o} = - \frac{-7.2}{12} = 0.6\] \[h_i = M \cdot h_o = 0.6 \cdot 4 \text{ cm} = 2.4 \text{ cm}\]

Atomic Structure

Atoms are the basic units of matter. Everything around us, including our bodies and the tools used in medical imaging, is made up of atoms. Knowing the structure of atoms helps us understand how radiation in medical imaging interacts with tissues and how machines detect or manipulate these interactions.

Basic Parts of the Atom

While there are over a hundred types of atoms in the universe (that we’ve discovered so far), each atom has three main subatomic particles: protons, neutrons, and electrons. The number of these particles in each atom determines the properties of the atom such as the identity, charge, or mass. This chart explains the key characteristics of each one:

Particle	Charge	Location	Relative Mass
proton	positive (\(+1\))	in the nucleus	\(\approx 1\) amu
neutron	neutral (\(0\))	in the nucleus	\(\approx 1\) amu
electron	negative (\(-1\))	orbiting the nucleus	\(\approx 0.0005\) amu

Binding Energy

The binding energy is the energy needed to hold the nucleus of an atom together. Without it, the positively charged protons would repel each other and break the nucleus apart. The larger the binding energy, the more stable the nucleus.

This is the formula for finding the binding energy of an atom:

\[E = mc^2\]

where \(E\) is the binding energy, \(m\) is the mass defect (difference between the total mass of separate nucleons and the mass of the nucleus), and \(c\) is the speed of light.

Note: While \(E = mc^2\) is how it is commonly written, you may also see it written as \(E = \Delta mc^2\). Just know they mean the same thing (\(\Delta m=m\)).

Stable Atoms

A stable atom is one whose nucleus doesn’t change or break apart over time. Stable atoms must have a balanced ratio of protons to neutrons. As a general rule, they will have the same number of both particles, though they may have slightly more neutrons. Too many or too few neutrons can make the nucleus unstable, which can lead to radioactive decay.

Electricity

Electricity is the flow of electrons, especially the valence electrons, which are the electrons in the outermost energy level of an atom. If valence electrons are loosely held, they can move from one atom to another. That’s what allows electric current to flow in metals, wires, and imaging machines. Materials that let electrons move easily are conductors (e.g., copper), and materials that don’t are insulators (e.g., rubber and glass).

Electrical charges are measured in coulombs (\(C\)).

Coulomb’s Law

Coulomb’s law states that the magnitude of the electrostatic force (\(F\); measured in newtons) between two point charges (\(q_1\) and \(q_2\)) is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance (\(r\)) between them. This is the formula:

\[F = k\frac{q_1 q_2}{r^2}\]

Note: In this formula, \(k\) is Coulomb’s constant of proportionality:

\[k = \frac{1}{4 \pi \epsilon_0} = 8.99 \times 10^9 \text{ N}\text{·} \text{m}^2/\text{C}^2\]

where \(\epsilon_0\) is the permittivity of free space, a constant that describes how electric fields interact in a vacuum (\(8.85\times10^{-12} \text{ C/N\text{·}m}^2\)).

Let’s try an example problem with this formula.

What is the electrostatic force between two charges, each with \(q = 2.0 \times 10^{-6} \text{ C}\), placed \(0.03\text{ m}\) apart?

Solution

We’re going to use Coulomb’s law:

\[F = \frac{k q_1 q_2}{r^2}\]

Now, substituting the given values, we have:

\[F = \frac{(8.99 \times 10^9)(2.0 \times 10^{-6})(2.0 \times 10^{-6})}{(0.03)^2}\]

Next, we’ll multiply the charges, remembering that when multiplying powers of \(10\), we add the exponents.

\[(2.0 \times 10^{-6})(2.0 \times 10^{-6}) = 4.0 \times 10 ^{-12}\]

The next step is multiplying by Coulomb’s constant. Remember to multiply the coefficients normally, then add the exponents for the powers of \(10\):

\[(8.99\times10^9)(4.0\times10^{-12})=25.96\times10^{-3}=0.03596\]

Now, we’ll square the distance:

\[(0.03)^2=0.0009\]

Finally, we divide the numerator by the denominator and have our answer:

\[F=\frac{0.03596}{0.0009}=39.96 \text{ N}\]

Electric Fields

An electric field is the region around a charged object where other charges feel a force. If you place a positive test charge on this field, it will experience a push or pull depending on the direction of the field. Electric fields, measured in newtons per coulomb (\(\text{N/C}\)), point away from positive charges and toward negative charges.

There are two formulas for finding the electric field:

\[E = \frac{F}{q} \text{ or }E = \frac{kQ}{r^2}\]

where \(E\) is the electric field, \(F\) is the electric force acting on the test charge, \(q\) is the test charge, \(k\) is Coulomb’s constant of proportionality, \(Q\) is the source charge, and \(r\) is the distance from the source.

Let’s do an example problem.

What is the electric field \(0.05\text{ m}\) away from a charge of \(3.0 \times 10^{-6} \text{ C}\)?

Solution

\[E = \frac{kQ}{r^2} = \frac{(8.99 \times 10^9)(3.0 \times 10^{-6})}{(0.05)^2} \approx 1.08 \times 10^7 \text{ N/C}\]

Circuits

A circuit is a complete loop that allows electric current to flow. It needs a power source (like a battery), a path for electrons (wires), and a device to use the electricity (like a light bulb or resistor). If the loop is broken at any point, the current stops flowing.

Ohm’s Law

Ohm’s law is the basic rule for how electricity behaves in a circuit. It shows the relationship between these variables:

• voltage (\(V\))—the push that makes electrons move; measured in volts (\(\text{V}\))
• current (\(I\))—how many electrons are flowing past a given point; measured in amperes (\(\text{A}\))
• resistance (\(R\))—how much a material resists the flow of current; measured in ohms (\(\Omega\))

This formula shows how these variables relate:

\[V = IR\]

Circuit Types

There are two main types of circuits: series and parallel. You must know which type of circuit you are working with to be able to accurately calculate the variables discussed above. Their properties are summarized in this box:

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In a specific situation, you may be required to figure out one of the three variables given the other two. Let’s try an example problem.

You have three resistors (\(2\, \Omega\), \(3\, \Omega\), and \(5\, \Omega\)) in series in a circuit where the total current is \(0.5\text{ A}\). What is the total voltage?

We are told these resistors are in a series, so that determines how we add them together. We start by finding the total resistance:

\[R_t = R_1 + R_2 + R_3 = 2 + 3 + 5 = 10\, \Omega\]

Now we can find the total voltage:

\[V_t = I_t \cdot R_t = (0.5 \text{ A})(10\, \Omega) = 5 \text{ V}\]

Magnetism and Electricity

Electricity and magnetism are closely connected; they are both part of a bigger force called electromagnetism. When electric current flows through a wire, it creates a magnetic field, and when a magnet moves near a wire, it can generate electric current (this is how generators work).

Magnetic Poles

Magnets always have two poles: a north pole and a south pole. Like poles repel each other, and opposite poles attract.

The strength of a magnetic field is measured in teslas (\(\bf\text{T}\)). A strong fridge magnet is about \(0.01\text{ T}\), while an MRI machine can be \(1.5\text{ T}\) to \(3.0\text{ T}\) or higher.

Electromagnetism

Electromagnetism is the interaction between electricity and magnetism. When current flows through a conductor (like copper wire), it generates a magnetic field around the wire.

Electromagnetic induction is the process of using a magnetic field to make electricity. If you move a magnet near a wire, a current is induced in the wire. This is how generators and transformers work.

The direction of the magnetic field created by a current flowing through a straight wire can be found by applying Fleming’s right-hand rule, as illustrated below:

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When the thumb points in the direction of the current (in this illustration, up), the fingers curl around the wire in the same direction of the magnetic field.