Physics at a Glance

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Displacement is a vector quantity with a magnitude and direction. It is a measure of the distance from the starting point to the end point in a straight line.

Distance is a measure along the path taken. It is a scalar quantity, having only magnitude. The odometer in a car measures distance.

Velocity is a vector quantity having magnitude and direction. In a straight line motion, the direction is denoted with a plus(+) or minus (−) sign. Unlike mathematics, signs in physics usually denote direction, not positive or negative numbers.

Acceleration is a vector quantity.

Accuracy is a measure of how close an experimental calculation is to a correct value such as the density of a substance.

Precision is a measure of how close a series of trials are to one another. For instance, a balance may be precise to ±0.01 grams. Every time the same object is ‘weighed’ on the balance the mass, the value of the mass will be the same plus or minus 0.01 grams.

To determine the accuracy of an experimental value, one needs to compare the experimental value to the actual value. An absolute error is the difference between the two numbers. The relative error is the per cent difference of the two values compared to the actual value.

When there is no actual value known, one can compare each of the experimental measurements to the average of all the trials. This is called deviation. The absolute deviation is the difference between a trial and the mean value. The relative deviation is the per cent difference between a trial and the mean value.

This is a straight line whose slope is m and whose y intercept is b.

An inverse relationship forms a hyperbola. That is a shape whose asymptotes are parallel to the x and y axes.

A parabola is formed when an equation is in the form .

Rules for Significant Figures

1] All non-zero digits are significant. (1,2,3,4,5,6,7,8,9)

2] All zeroes between two non-zero digits are significant. (1001 has 4 sig. figs.)

3] All zeroes to the left of the first non-zero digit are NOT significant. (0.00012 has 2 sig. figs.)

4] All zeroes to the right of the last non-zero digit and to the left of an understood decimal point are NOT significant. (2100 has 2 sig. figs.)

5] All zeroes to the right of the last non-zero digit where a decimal point is written are significant. (2100. or 0.002100, both have 4 sig. figs.)

6] All digits in the coefficient of a number in scientific notation is significant. (3.11×10⁷ has 3 sig. figs.)

7] When rounding a number which ends exactly with a five (5), then round to the even preceding digit. This means that if the number preceding the five is an odd number, the number rounds up to the even number. If the number preceding the five is an even number, it remains the same. One always rounds to an even number. (2.35 becomes 2.4) (2.450 becomes 2.4)

8] When adding or subtracting measurements, round all the numbers to the same column as the least accurately measured value. Then add or subtract. (2.34 +1.234 + 3.4 = 2.3 +1.2 + 3.4 = 6.9)

9] When multiplying or dividing measurements, do the calculation first. Then round the answer to the same number of significant digits as the original values with the fewest significant digits. (17.28 ÷1.2 = 1.4)

Significant digits only pertain to measurements which have been made. Numerical constants can be considered exact values.

Counting numbers are not measurements and are considered exact.

Numerical constants like π should be written with one more digit than the best known other number which is given when solving an equation using π.

Scalar quantities have a magnitude only.

Vectors have a magnitude and a direction associated with them. Some directions are understood such as your weight which is a force pointing toward the center of the Earth.

Vectors are equal if they point in the same direction and have the same magnitude. Vectors can be moved from place to place as long as they keep the same magnitude and point in the same direction.

To add vectors, align them tail to head and the resultant is the line which runs from the tail of the first vector to the head of the last vector.

When two vectors are at right angles:

If there is no right angle, use the Law of Cosines which is the general rule from which the Pythagorean Theorem is derived.

Lower case a, b, and c are the respective sides of the triangle and A, B, and C are the angle across the triangle from sides a, b, and c.

We also may need to use the Law of Sines:

To add two vectors which are at right angles, use the Pythagorean Theorem.  The Pythagorean Theorem is a special case of the Law of Cosines.

The angle from the first vector to the resultant (answer) is

A single vector can be divided into two components along any axes which are at right angles like the x and y axes.

The slope of a graph which shows displacement vs. time gives the velocity.

The slope of a graph of velocity vs. time gives the acceleration.

The area under a velocity vs. time graph gives the displacement at final time.

The area under an acceleration vs. time graph gives the velocity at the final time.

Free fall is the acceleration due to gravity with no other forces acting. The acceleration of gravity is a special case and is denoted by the letter, g.

where the negative sign indicates down.

Normally, up is considered positive and down is considered negative. One notable exception is in the determination of weight where we know the direction is down. We use to determine the weight of an object since we know weight points downward.

1. (Galileo’s Law of Inertia) An object at rest will remain at rest and an object in motion will continue moving in a straight line at a constant speed UNLESS it is acted upon by an outside force.

2. (Law of Acceleration) The acceleration of an object is directly proportional to, and in the same direction as the net force applied to it. The acceleration is inversely proportional to the mass of the object.

3. (Law of Interaction) For every action there is an equal and opposite reaction.

Forces can only exist in pairs with one opposing the other.

The net force is what remains after vectorially adding all the forces being applied to the object in question.

Mass is the same everywhere while weight changes from place to place with the acceleration of gravity.

Apparent weight is what one feels. Standing on a bathroom scale gives your apparent weight. If one stood on a scale in an elevator, the apparent weight would change when starting or stopping due to a difference in the acceleration one feels. During motion on an elevator, one’s apparent weight is always greater at the bottom of the ride and less at the top of the ride.

Friction is a resistance to motion. Friction always acts to oppose motion. The force is anti-parallel to the direction of motion and perpendicular to the contact force. Friction may be static or kinetic. Static friction exists between an object and the surface when there is no motion along the surface. One can push against a heavy box on a rough surface without moving it. The force opposing the push is static friction. Kinetic friction is the opposing force when the object is moving across the surface.

The force with which the surface pushes against the object is the Normal Force. On a horizontal surface, the Normal Force is equal to and opposite the weight of the object.

The coefficient of friction (μ) is the ratio of the friction force to the Normal Force. For kinetic (or sliding) friction, . The friction force is sometimes written as f and the Normal force is sometimes written as N.

Since friction can never move an object, static friction may be smaller than the maximum value. . If one pushes against an object with a 12 newton force and the maximum static friction force is 20 newtons, friction will only equal the 12 newtons.

When a sky diver or any object falls through the atmosphere, there is air drag. As the speed of the object increases downward, the air friction force increases. Eventually the air drag force equals the weight of the person or object and there is no further increase in speed. This maximum speed is the terminal velocity of the person or object. When a sky diver "belly flops," this is about 125 mph. If a sky diver goes head (or feet) first there is less surface area (less friction) and the terminal velocity is about 150 mph.

A pendulum swings back and forth (or up and down) in a regular periodic manner.  This is called simple harmonic motion (SHM).  The period of the motion depends on the length of a string pendulum and the mass of a spring pendulum along with constants.

Forces in ropes, strings or the like are called tensions since one can only pull with a rope and not push.

Equilibrium is a state where an object is either at rest or in motion at a constant velocity. In either case, there is no acceleration.

This is the first condition of equilibrium.

The Equilibrant is equal to and opposite in direction to the resultant.

The path of a projectile is its trajectory. The horizontal velocity is constant. The horizontal velocity is independent of the vertical velocity. The vertical velocity is accelerated by gravity at . The vertical velocity is zero at the top of the path. When the projectile is fired on a level surface, the initial vertical velocity is equal to and opposite the vertical velocity at the end of the path. The object travels along a parabolic path.

The range along a horizontal surface is .

To travel in a circular path, an object must be accelerated toward the center of the circle. This is the centripetal acceleration.

Some people insist a force pushes them outward, the centrifugal force. This is a nonexistent force. Inertia makes one think there is an outward force.

A torque causes rotation. It is the product of the applied force and the lever arm which is the perpendicular (or shortest) distance from the force to the pivot point (fulcrum).

The second condition of equilibrium is .

1. All planets travel in ellipses.

2. An imaginary line from a planet to the sun sweeps out equal areas in equal times.

3. The square of the ration of the periods of any two planets orbiting the sun is equal to the cube of the ratio of their average distances from the sun.

Gravitational Mass

Momentum           Impulse

Impulse-Momentum Theorem

Conservation of Momentum

Perfectly inelastic collisions occur when two objects collide and stick together after the collision. This causes the greatest loss of mechanical energy into non-mechanical forms (mostly heat).

Inelastic collisions are those where kinetic energy is not conserved. Objects may rebound from one another, but lose energy.

Elastic collisions are those where both kinetic energy and momentum are conserved.

In an elastic collision there is only one possible answer for the velocities after the collision. These are:

Angles are measured in radians for angular motion. One radian is the angle subtended when the distance along the circumference is equal to the radius of the circle.

Tangential displacement, velocity and acceleration

Tangential quantities are measured tangent to the circular motion.

Moment of Inertia is the product of the mass of a small piece and the square of distance from the axis of rotation. For some solid shapes, the values are given below.

Torques

Angular Momentum

Work is done when a force is applied to an object moving it in the direction of the force

When work is done at an angle to the surface on which it is moved

Angular Work

Kinetic Energy

Gravitational Potential Energy

Spring Potential Energy

Hooke’s Law of Springs where k is the spring constant and the negative sign denotes that the spring pulls or pushes in the opposite direction.

Power

Rotational Kinetic Energy

Angular Power

Simple Machines

Archimedes’ Principle

The buoyant force on a submerged object is equal to the weight of the water the object displaces.

Pressure

Pressure is measured in newtons per square metre.

Pascal’s Principle

Pressure applied to fluid in a closed container is transmitted equally in all directions to every point of the fluid and to the walls of the container.

Fluid Pressure as a function of depth

Absolute Pressure equals the atmospheric pressure plus density times acceleration of gravity times the depth

Fluid Flow The rate of fluid flow changes with the cross-sectional area of the pipe through which it flows

Continuity Equation

Bernoulli’s Principle: The pressure in a fluid decreases as the fluid’s velocity increases.

The pressure in a fluid plus the kinetic energy per unit volume plus the gravitational potential energy per unit volume is constant.

Charles’ Law

Boyle’s Law

Gay-Lussac’s Law

Combined Gas Law

Ideal Gas Law where N is the number of molecules and k_B is Boltzman’s constant. k_B = 1.38×10 ²³ Joules per Kelvin.

The Ideal Gas Law can also be written as where n is the number of moles of the gas and R equals 8.31 Joules per (mole Kelvin). The pressure is in Newtons per square metre (Pascals) and the volume is in cubic metres (steres).

Avogadro’s Number

Temperature Scales

Temperature is a measure of the average kinetic energy of all the molecules in the substance which one is measuring.

Materials bend and stretch. The stress relates the force to the cross-sectional area of the object. Strain is the ratio of the change in length to the original length.

The length can also change. See the Heat section.

Heat is a measure of the total kinetic energy of all the molecules in the substance which one is measuring. Heat is measured in Joules or calories.

Internal energy is the energy associated with the motion of the molecules in a substance.

is the Law of Conservation of Energy including the heat in the substances.

The amount of heat required to raise the temperature of a substance 1.000 K, is its specific heat capacity. where Q is the amount of heat in Joules.

When a substance changes phases, from a gas to a liquid or a liquid to a solid, or visa versa, heat energy is used to break or unite bonds in the substance. This occurs without a temperature change in the substance and is called latent heat. The phase change of liquid to solid is called fusion and thus there is a latent heat of fusion. The change from a liquid to a gas is called vaporization and thus there is a latent heat of vaporization. . Note there is no temperature in the formula.

Heat is transferred by conduction, convection and radiation. The rate of transfer depends on the difference in temperature. The larger the difference, the faster the transfer of heat. Conduction is the transfer from molecules to molecule. Convection is the transfer by a heated fluid (liquids and gases) rising and taking the heat away. Radiation is the transfer of heat by waves which can happen through a vacuum as the sun’s energy is transferred to the earth.

Sublimation is the process of changing a gas directly into its solid phase or a solid directly going to the gaseous phase. Dry Ice (solid carbon dioxide does not change into a liquid. It sublimates directly from solid to gas.

Materials expand when heated and contract when cooled.

The coefficient of expansion varies with the material and also varies slightly with the temperature.

The volume of a substance can also vary with the temperature.

Sound travels through a medium. Sound is a wave. Waves transmit energy from one place to another without displacing any mass. The waveform moves carrying the energy.

Waves can be either longitudinal or transverse. The particles in a transverse wave move perpendicular to the direction of motion. This occurs when a wave moves along a rope. The rope remains in the same position before and after the wave passes. The particles of rope move up and down as the wave passes any given point. Transverse waves can travel through solids only.

The particles in a longitudinal wave move parallel to the direction of motion. An example would be when one squeezes coils of a spring together and then release the coils. The coils move apart squeezing the coils in front of them together and as they expand they squeeze the coils in front of them. This continues until the energy dissipates. Longitudinal waves can travel through solids, liquids, or gases.

Any wave can reflect, refract or diffract. Reflection occurs when a wave bounces from a surface. Transverse waves reflect 180 out of phase. The angle of incidence equals the angle of reflection, . Refraction is the bending of wave as it crosses from one medium to another. Waves also bend when passing the edge of a barrier, such as when sound travels around the edge of a doorway. This is called diffraction.

Sound is a longitudinal wave. The air molecules in one’s throat are compressed by the up and down motion of the vocal cords. The compressed air moves out of the mouth compressing the air in front until it reaches another person’s ear. It then pushes against the ear drum and causes it to move in the same fashion as the vocal cords were moving.

Frequency is the number of waves passing a point in a unit of time, usually one second. The inverse (or reciprocal) of the frequency is the period of the wave.

Sound resonates (as does any wave) under certain conditions. When a sound wave travels through a tube, either open of closed at one end, it may resonate if the wavelength fits in the tube. If the sound wave travels down a closed tube, bounces off the bottom and reaches the top just as a new wave enters the tube, the amplitude (or height) of the wave increases making the sound louder. This is resonance.

The speed of speed depends on the air temperature (and air pressure). Since air pressure on the surface of the earth does not vary greatly, we can use a simple formula to determine the speed of sound.

The Doppler Effect is the change in the frequency of a wave as it is emitted or reflected from a moving object. This effect is used by the police to determine the speed of an automobile.

Whenever the listener is moving toward a source of sound or when the source of the sound in moving toward the listener, the frequency heard is higher than the frequency emitted by the source. Whenever the listener is moving away from the source or the source is moving away from the listener, the frequency heard by the listener is lower than the frequency emitted by the source.

LF means the listener is in front and the source of the sound is approaching the listener.

LB means the listener is behind the source and the source is moving away from the listener.

LC means the listener is closing in on the source. The listener is moving toward a stationary source.

LO means the listener is opening the distance from a stationary source. The listener is moving away from a stationary source.

V is the speed of sound. V_S is the speed of the moving source. V_L is the speed of the moving listener.

When both the listener and the source are moving, it is easiest to do the problem twice. First do the problem as if the source were stationary, then use that answer and do the problems as if the listener when stationary. This could also be done in the reverse order.

Strings (as guitar strings) can vibrate to create sounds. The frequency emitted by the vibrating strings varies with the length of the string, the density of the string, the tension in the string, and the diameter of the string.

If more than one of these factors change, one can calculate the change for one factor, then use that factor for the next change and continue as many times as necessary.

Sound intensity is the power per unit area, , watts per square metre. There is a lower limit to the sound intensity which can be detected by the human ear. A minimum amount of energy per second is needed to vibrate the eardrum. That intensity is the threshold of hearing, . This value is approximate and may vary slightly from person to person.

Sound level is measured in bels. A bel is a large unit so the decibel is a more convenient unit. The sound level varies exponentially. The sound level is written based on logarithms.

The sound intensity (I) is a sum of all the separate sound intensities. Sound levels are measured in two ways. The can be measured instantaneously, giving a maximum or minimum value, but more commonly they are averaged over a short period of time. Meters which do this are used by police is communities which have laws governing the sound level at parties and businesses.

At approximately 120 decibels, people no longer feel the intensity of sound, but rather experience pain. Again, this varies from person to person. Some people feel pain at 118 decibels and others at 121 or 122 decibels.

Hearing loss occurs when one is exposed to levels above 90 decibels. Concerts have been measured at above 125 decibels near the speakers. People attending the concerts experience a partial deafness when they leave. Eventually, approximately 99% of their original hearing returns, but 1% is lost forever. When they go to another concert 1% of their remaining hearing would be lost and so on.

Optics is the study of light rays. When light hits a mirror or specular surface, it reflects so that the angle of incidence equals the angle of reflection.

A plane mirror always shows an image which is upright. An upright image is a virtual image. Light rays do not pass through the location of the image. The image from a plane mirror is always behind the mirror. The light rays do not cross behind the mirror.

A real image is formed when light rays cross. Real images can be formed on a screen. Real images are always inverted.

Mirrors

Mirrors can be curved. There are spherical and parabolic mirrors.

Spherical mirrors have a focal point which is half the radius of curvature. Spherical mirrors are used in flood lights and the high beams of auto headlights because the light rays far from the principle axis will spread out farther.

A parabolic mirror is shaped like a parabola in three dimensions. A parabolic mirror will focus all the light rays parallel to the principle axis. A parabolic mirror is used in spot lights and the low beams of car headlights.

Mirrors in reflecting telescopes are all spherical mirror which have a very large radius of curvature.

Spherical mirrors may be concave or convex. When one looks into a concave mirror, the center of the mirror is farther away than the edges. A convex mirror is curved such that the center of the mirror is closer than the edges of the mirror.

Convex mirrors are used in stores to see a larger area. The images all appear smaller than if a plane mirror were used. The convex mirror is also used on the passenger side of automobiles to view a larger part of the road and say "Objects in the mirror are closer then they appear."

Concave mirrors are used for cosmetic and shaving mirrors so that one’s face appears larger than normal.

A convex mirror only produces virtual images which are upright and behind the mirror. The images always are smaller than the object.

Concave mirrors produce virtual images when the object is closer than the focal length from the mirror. These images always are enlarged and are behind the mirror. When the object is between the focal point and the center of curvature, the image is real, inverted and farther from the mirror than the center of curvature. If the object is at the center of curvature, the image is real and inverted and is also located at the center of curvature. If the object is located beyond the center of curvature, then the image is real and inverted and lies between the focal point and the center of curvature.

The focal point of a convex mirror is located behind the mirror and distances behind the mirror are negative. The focal point of a concave mirror is in front of the mirror and distance in front of the mirror are considered positive.

where f is the focal length of the mirror, d_o is the distance of the object, and d_i is the distance of the image. Recall that distance is positive in front of the mirror and negative behind the mirror.

where d_i is the distance of the image and d_o is the distance of the object. If the magnification is positive, the image is virtual and upright. If the magnification is negative the image is real and inverted. One can also think of the magnification as being an absolute value and determine if the image is real or virtual from the distances in front of or behind the mirror.

When light rays cross a boundary the light ray is bent or refracted causing it to change direction. When a light passes through a window at some angle. The ray exits the opposite side of the glass at the same angle, but slightly displaced from the original direction.

Wilebrod Snell described how the angle changed when the light ray passed into a different medium. Snell’s Law describes how the angle changes, . The index of refraction, n, is the ratio of the speed of light in a vacuum to the speed of light in the substance. The index of refraction is always greater than or equal to 1.000. The index of refraction is a measure of the optical density of the substance. The index of refraction of air is 1.0003, so close to 1.000 that it is usually called 1.000 in problems.

In a lens, the glass is curved so that the opposite sides of the lens are not parallel except at the center of the lens. This causes the light to bend twice as it passes through the lens. Depending on the shape of the lens, the light rays can converge to a point or diverge away from a single point.

Converging lenses are also called convex lenses since they are thicker at the center than at the edges. Diverging lenses are also called concave lenses since they are thinner at the center than at the edges.

There are three types of convex lenses and three types of concave lenses.

For a converging lens, one light ray from the object travels parallel to the vertical center of the lens and then travels through the focus on the opposite side of the lens. Another light ray from the same point travels through the geometric center of the lens in a straight line. A third ray travels from the same point through the focus on the same side of the lens to the vertical center of the lens and then travels parallel to the principle axis. All the light rays cross at the same point and form the real image (inverted) on the opposite side of the lens.

An object at the center of curvature forms an image at the center of curvature on the opposite side of the lens. An object between the focus and the center of curvature forms a real image beyond the center of curvature on the opposite side of the lens. An object beyond the center of curvature forms a real image between the focus and the center of curvature on the opposite side of the lens.

No image is formed when the object is placed at the focus (focal point).

When an object is closer to the lens than the focus, a virtual image is formed. A virtual image is on the same side of the lens as the object and is enlarged and upright. This is a magnifying glass.

All images formed by a diverging (concave) lens are virtual. They are all upright and smaller than the object.

A line ray is drawn parallel to the principle axis to the vertical center of the lens. A line drawn from the focus to the point where the ray intersects the vertical center of the lens continues beyond the lens diverging from the principle axis. A second ray is drawn from the object through the geometric center of the lens. The image forms where the two rays cross between the focus and the lens.

where f is the focal length of the lens, d_o is the distance of the object, and d_i is the distance of the image. Recall that distance is positive on the opposite side of the lens and is negative on the same side of the lens. The focal length is positive for convex (converging) lenses and is negative for concave (diverging) lenses.

where d_i is the distance of the image and d_o is the distance of the object. If the magnification is positive, the image is virtual and upright. If the magnification is negative the image is real and inverted. One can also think of the magnification as being an absolute value and determine if the image is real or virtual from the distances on the same side of or on the opposite side of the lens.

Real lenses depend on the index of refraction of the glass being used as well as the curvature of both sides of the lens.

If the radius of the far side of the lens is measured and center is located on near side, the radius is positive. If the center is on the far side of the lens, the radius is negative. Then when evaluating the focal length, a negative focal length is a concave (diverging) lens and a positive value is a convex (converging) lens.

Electric charge comes from protons and electrons. Electrons have a negative charge of 1.602×10 ¹⁹ Coulombs. Protons have a positive charge equal to the charge on an electron. Electrons may be added to or removed from an object since they orbit the nucleus of an atom where the protons reside. Objects which become negatively charged have an excess of electrons. Objects that are positively charged have fewer electrons.

Objects with opposite charges attract one another. Objects with like charges repel one another.

Charles August de Coulomb determined the law that finds the amount of force between static charges. The constant, k, is equal to . The unit of charge was named for him. It was Benjamin Franklin who gave us the idea of positive and negative charges and the assumption that charge moved from positive to negative. We now understand that it is the negative electrons that move, but we still use the idea that electricity will travel from a positive charge toward a negative.

At the University of Leyden in Germany electricity was first studied in detail. It was found that one could store electricity in a Leyden jar. This was a device which consisted of a glass jar with silver foil on the inside and the outside. A rubber stopper has a metal rod extending through it and a small chain which hung down to touch the foil on the inner surface.

Rubbing a rubber rod with a piece of fur moves electrons (negative charges) onto the rod. These can be placed in the Leyden jar by touching the metal rod. The electrons move until they are evenly distributed between the rod and the jar. This can be done over and over until the charges are nearly even. The electrons on the inner surface repel electrons on the outer surface and attract the remaining positive charges.

This can also be done by rubbing a glass rod with a piece of silk. However the glass rod gives away electrons leaving a positive charge on the rod.

An electroscope consists of a metal rod extending into a glass jar which has two pieces of metal foil hanging from it. Originally these pieces were gold foil which is extremely thin. A charge accumulating on the pieces of foil would cause them to repel. Whenever the foil separated, one knew that a charge had been placed on the foil, but one could not determine if it were positive or negative.

If a charged rod is brought near the electroscope, the leaves would separate because the charge on the rod would repel a similar charge on the metal rod causing those charges to accumulate on the leaves, repelling each other. One could then touch the metal rod and the leaves would come back together. When the rod is removed from the vicinity and the hand is moved away the leaves would again move apart. This is charging by induction.

Electric fields are the ratio of the force which Coulomb measured and the charge causing the force. The force decreases with distance. Coulomb’s Law can be substituted into this equation to get a different version. Michael Faraday viewed the electric field as lines that emanated (pointed away) from a positive charge and pointed into a negative charge.

Alessandro Volta built the first electric pile, a series of pieces of metal separated by pieces of cloth soaked with a weak acid. This was the first battery and a means to create moving electric charge.

When electricity flows through a wire, one can make an analogy with water flowing through a pipe. The water is similar to the electric charge. Volts are similar to the pressure pushing the water through the pipe. Amperes (named for Andre Ampere) are similar to the rate at which water flows through the pipe.

Electric current is the charge moved per second (Coulombs per second).

Most of our interest in electricity is that of electric current. Georg Simon Ohm investigated electricity moving through wires. He found that there was one other important quantity which was missing: electrical resistance. This resistance is similar to the size of the water pipe described above. The smaller the pipe the more difficult it is for the water to get through the pipe. The larger the pipe the lower the resistance and the easier it is for water to move through the pipe. The same is true for electricity moving through wires. Large diameter wire can carry greater amounts of electricity.

Ohm found that the voltage is equal to the product of the current and the resistance. The unit of resistance is named for Ohm and has the symbol, . Ohm’s law is written as . He described voltage as the work done to move the charge divided by the charge. This is measured in Joules per Coulomb.

Electric power is measured in watts, the same as power in mechanics. Watts are Joules per second, so electric power is .

When one buys electricity, they are paying not for power but for energy. Energy is power multiplied by the time. The electric company charges by kilowatt-hours. One kilowatt-hour is equivalent to 3,600,000 Joules of energy. The average apartment uses about 300 kilowatt-hours per month while a house uses about double that amount.

One can determine the cost of the electricity by multiplying the cost per kilowatt-hour by the charge. Electric companies always round the charge up. With the high cost of fuel to make the electricity, there is also a surcharge for the cost of the fuel added to the cost of the electricity.

Electric currents flow through two types of circuits, series and parallel. A series circuit is set up so that the electricity leaving a source has to travel through every object in the circuit before returning to the source. All the current in a series circuit has to pass through every object. The voltage leaving the power source drops off as it passes through each object and is zero when it returns to the power source. The sum of the voltage drops through each object will add up to the voltage leaving the power source. The total resistance of the circuit is equivalent to the sum of all the resistances.

In a parallel circuit, the current leaving the power source is distributed among the resistances. The voltage is the same everywhere in the circuit. The difference allows changes the equivalent resistance of the entire circuit. The total resistance is less than the smallest resistance in the circuit.

Some circuits are combinations of series and parallel circuits. The rules for each are the same, but only in the parts that are in series or the parts that are in parallel. These circuits are usually simplified by combining the parts until there is a circuit with only a single resistor. A sample is shown below.

Resistances R2 and R3 are combined to make resistor R6. 10+5=15Ω.

Resistances R4 and R6 are combined to make resistor R7. Then R1, R7 and R5 are combined to make R, the equivalent resistance for the entire circuit. R=8+10+7=25Ω.

The current leaving the battery is

Hans Christian Oersted noticed that a compass needle is deflected when it is brought near a wire carrying a direct current. He also noticed that the compass needle lined up perpendicular to the wire to the left or right above the wire. Then when the compass was placed below the wire, it lined up to the right or left (opposite) the direction above the wire. He determined that while a current flowed through the wire, a magnetic field was created circling the wire by the right-hand rule. The right-hand rule states that when one grabs a wire with their right hand such that the thumb points in the direction of the positive current, the fingers wrap around the wire in the direction of the magnetic field.

Later, it was noticed that moving a magnet near a wire caused a current to flow in the wire. This was the beginning of using electricity to make a motor and using magnets to create electricity in a generator.