# Ohm's Law Explained

**Four essential attributes** form the core of electron movement:

- Power to do work (P)
- Current Intensity (I)
- Electromotive force (E)
- Resistance (R)

**One way to visualize these attributes is to imagine electrons in copper wire.** If one end of the wire has an excess of electrons, the electrons will exert a force, moving toward the end with a lack of electrons. Thus, electricity begins with a certain Electromotive force, a difference in potential that "pushes" electrons in one direction.

**Electromotive force**, however, encounters some difficulty in moving electrons freely in every situation. Some materials resist electron movement more or less than others. This opposition to electromotive force is called Resistance...it actually limits the number of electrons that can be "pushed".

**The combined effect of resistance upon electromotive force** results in a specific number of electrons actually flowing. This flow of current is called Current Intensity, or simply current.

**Finally, as current intensity flows in our wire**, it generates heat, which is a form of energy, or the power to do work. We can use that power to ignite a cigarette, illuminate a light bulb, or turn a motor.

**The four attributes of electricity** can be remembered easily with the word, "PIER":

- P: Power
- I: Intensity (current flow)
- E: Electromotive force
- R: Resistance

**Each of the four attributes is measured differently:**

- Watts (measures power)
- Amperes (measures intensity, or current flow)
- Volts (measures electromotive force
- Ohms (measures resistance

**It is interesting to try to imagine** the number of electrons represented by the words, "One ampere":

### 6,241,509,480,000,000,000 electrons flowing each second!

**The relationship between these attributes** can be expressed with two mathematical equations:

## I = E x R

## P = I x E

**The first equation** says that Current Intensity (I) is the result of Electromotive force (E), controlled by Resistance (R). **One ampere** of current intensity results from **one volt** of electromotive force in a circuit with **one ohm** of resistance.

**The second equation** shows that Power (P), or the rate at which work is done or heat generated, is the result of current Intensity (I) driven by Electromotive force (E). **One watt** of power is the result of **one ampere** of current intensity driven by **one volt** of electromotive force.

**Our first equation can be placed on a chart**, under the column headed "I":

P | I | E | R |
---|---|---|---|

- | I = E x R | ||

**The equation is placed in the "I" column** because, as it is written now, it solves for current intensity. There is a dash in the first cell under the column "P" because this equation does not include any factor for power. Now we will concentrate on rearranging our first equation to solve for the remaining attributes.

**Basic rules of mathematics allow us to rearrange equations** as long as the original relationship remains equal. Dividing both sides of the equation by R results in a new form of the original relationship:

## I = E x R

## I/R = E x R/R

## I/R = E x 1

**Simplifying the result** and flipping sides gives us an equation that solves for E:

## E = I/R

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | |

**Next, starting again with the original equation**, we can divide both side by E to solve for R:

## I = E x R

## I/E = E/E x R

## I/E = 1 x R

## R = I/E

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

**Thus far, using just one equation, we've derived three different ways** to mathematically express the relationship between Power, Intensity of current, Electromotive force, and Resistance. Now we can start with the next equation, the Power Equation:

## P = I x E

**It can be directly placed in the second row**, under the column headed "P", because it solves for Power:

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | |||

**Dividing both sides of the equation** by E results in:

## P/E = I x E/E

**Simplifying and flipping sides gives us a new equation**, solving for I:

## I = P/E

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | ||

**Starting again with the Power Equation**, and dividing both sides by I, creates a new equation solving for E:

## P = I x E

## P/I = I/I x E

## P/I = 1 x E

## E = P/I

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

A dash is placed in the last column of the second row because this equation has no factor for Resistance.

**We've used both equations, rearranging them to produce six different ways** to mathematically express the relationship between Power, current Intensity, Electromotive force, and Resistance. Or, to put it in terms of actual units of measurement, we've developed six different equations to calculate watts, amperes, volts, and ohms in a circuit. If we know two of the factors, we can calculate the other two. Amazing!

**But wait...there's more!**

**We can substitute one of the equations for part of another equation**, creating an entirely new equation:

**We know that P = I x E, and we know that I = E x R**, so we can substitute E x R for I:

## P = (E x R) x E

This is the same as saying:

## P = E x E x R,

Which is the same as saying:

## P = E^{2} x R

**We now have a new expression for P**, using just the factors of E and R:

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | |||

**If P = E ^{2} x R**, then dividing both sides by R will give us an equation solving for E:

## P/R = E^{2} x R/R

## P/R = E^{2} x 1

## P/R = E^{2}

**If both sides of the equation are equal** (and they are), then the square roots of each side are equal (and they are)!

## E = √(P/R)

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | - | E = √(P/R) | |

Again, a dash is in the third row under the column headed "I" because this equation has no factor for current intensity.

**To use this equation to solve for R**, divide both sides by E^{2}:

## P/E^{2} = E^{2}/E^{2} x R

## P/E^{2} = 1 x R

## R = P/E^{2}

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | - | E = √(P/R) | R = P/E^{2} |

**We've exhausted the possiblities of this equation**, so let's return to our two original equations and see what can be substituted anew:

## P = I x E and E = I x R

Substitute I x R for E:

## P = I x (I x R)

Simplify:

## P = I^{2} x R

**This equation can be placed in the last row** under the column headed "P":

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | - | E = √(P/R) | R = P/E^{2} |

P = I^{2} x R |

**Solve for I by dividing both sides by R**:

## P/R = I^{2} x R/R

## P/R = I^{2} x 1

## √I^{2} = √P/R

## I= √P/R

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | - | E = √(P/R) | R = P/E^{2} |

P = I^{2} x R | I= √P/R | - |

A dash in the last row under the column headed "E" shows that this equation has no factor for electromotive force.

Solve for R by dividing both sides by I^{2}:

## P/I^{2} = I^{2}/I^{2} x R

## P/I^{2} = 1 x R

## R = P/I^{2}

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | - | E = √(P/R) | R = P/E^{2} |

P = I^{2} x E | I= √P/E | - | R = P/I^{2} |

**Each of our four factors has three different equations**, for a total of 12 equations derived from our original two.

P | I | E | R |
---|---|---|---|

- | I = E x R | E = I/R | R = I/E |

P = I x E | I = P/E | E = P/I | - |

P = E^{2} x R | - | E = √(P/R) | R = P/E^{2} |

P = I^{2} x E | I= √P/E | - | R = P/I^{2} |

**Simply amazing!**

This work is licensed under a Creative Commons Attribution 3.0 Unported License.