Order of Magnitude

Examples of Order of Magnitude in the following topics:

• Order of Magnitude Calculations

  • An order of magnitude is the class of scale of any amount in which each class contains values of a fixed ratio to the class preceding it.
  • An order of magnitude is the class of scale of any amount in which each class contains values of a fixed ratio to the class preceding it.
  • Such differences in order of magnitude can be measured on the logarithmic scale in "decades," or factors of ten.
  • The order of magnitude of a physical quantity is its magnitude in powers of ten when the physical quantity is expressed in powers of ten with one digit to the left of the decimal.
  • If they differ by two orders of magnitude, they differ by a factor of about 100.

• A Practical Aside - Orders of Magnitude

  • One of the most important tools that a card-carrying astrophysics has is the order of magnitude estimate.
  • The order of magnitude estimate combines the lack of rigor of dimensional analysis with the lack of accuracy of keeping track of only the exponents; this makes multiplication in your head easier!
  • The first part of the tool is the knowledge of the various constants of nature in c.g.s units but you only need to keep the exponent in your head.
  • A glance at the following table shows that some of the physical constants are easier to remember than others, but one can exploit the relationships between them a remember only a few key numbers to obtain the the rest.

• Scientific Notation

  • In order to go between scientific notation and decimals, the decimal point is moved the number of spaces indicated by the exponent.
  • Another way of writing this expression, as seen on calculators and computer programs, is to use E to represent "times ten to the power of."
  • Scientific notation enables comparisons between orders of magnitude.
  • As seen above, scientific notation uses base 10, and if a number is an order of magnitude greater than another, it is 10 times larger.
  • For example, 4.759 x 106 is 3 orders of magnitude bigger than 5 x 103; it is 8 orders of magnitude bigger than 2.56 x 10-2.

• Problems

  • Let's take a one-dimensional model of this.
  • Calculate the energy and wavelength of the hyperfine transition of the hydrogen atom.
  • We are looking for an order of magnitude estimate of the wavelength.
  • Calculate the energy and wavelength of the transition of hydrogen with the spin of the electron and proton aligned to antialigned.
  • Calculate the ionized fraction of pure hydrogen as a function of the density for a fixed temperature.

• Multiplying Vectors by a Scalar

  • Multiplying a vector by a scalar changes the magnitude of the vector but not the direction.
  • This will result in a new vector with the same direction but the product of the two magnitudes.
  • For example, if you have a vector A with a certain magnitude and direction, multiplying it by a scalar a with magnitude 0.5 will give a new vector with a magnitude of half the original.
  • For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds.
  • In order to make this conversion from magnitudes to velocity, one must multiply the unit vector in a particular direction by these scalars.

• Scalars vs. Vectors

  • Vectors require two pieces of information: the magnitude and direction .
  • The magnitude of a vector is a number for comparing one vector to another.
  • Each of these quantities has both a magnitude (how far or how fast) and a direction.
  • In order to specify a direction, there must be something to which the direction is relative.
  • Ideas about magnitude and direction are introduced and examples of both vectors and scalars are given.

• Addition of Velocities

  • In order to calculate the velocity that the object is moving relative to earth, it is helpful to remember that velocity is a vector.
  • In order to analytically add these vectors, you need to remember the relationship between the magnitude and direction of the vector and its components on the x and y axis of the coordinate system:
  • The first two equations are for when the magnitude and direction are known, but you are looking for the components.
  • The last two equations are for when the components are known, and you are looking for the magnitude and direction.
  • The magnitude of the observed velocity from the shore is the square root sum of the squared velocity of the boat and the squared velocity of the river.

• Forces in Two Dimensions

  • For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result.
  • For example, if you know that two people are pulling on the same rope with known magnitudes of force but you do not know which direction either person is pulling, it is impossible to determine what the acceleration of the rope will be.
  • In this simple one-dimensional example, without knowing the direction of the forces it is impossible to decide whether the net force is the result of adding the two force magnitudes or subtracting one from the other.
  • The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.
  • Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the net force.

• An Astronomical Aside: Magnitudes

  • Astronomers typically speak about the flux of an object in terms of magnitudes.
  • A magnitude is generally defined as
  • Pogson empirically determined the value of "2.5" by comparing the magnitudes of prominent observers of the 1800's.
  • It is remarkably close to $\ln 10 \approx 2.3$, so a change in magnitude of 0.1 is about a ten-percent change in flux.
  • Two of the standard conventions are the "Vega" convention which states that the magnitude of the star Vega regardless of the function $g(\nu)$ is zero; all of Vega's colours are zero.

• Introduction to Scalars and Vectors

  • A vector is any quantity that has both magnitude and direction, whereas a scalar has only magnitude.
  • Whereas displacement is defined by both direction and magnitude, distance is defined by magnitude alone.
  • Distance is an example of a scalar quantity.
  • A vector is any quantity with both magnitude and direction.
  • Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down.