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Voltage
Voltage is the change in electric potential (meaning potential energy per unit charge) between two positions. The voltage is always measured between two points, for example between the positive and negative ends of a battery, or between a wire and ground. It is measured in volts.
It was named after an Italian physicist Alessandro Volta who made the first chemical battery. Voltage is also called electric tension.
The voltage can be seen as the pressure on the electrons to move out of the source. It is directly proportional to the pressure exerted on the electrons. In other words, the higher the voltage, the higher the pressure. For example, a battery of 3 volt will exert pressure on the electrons twice as hard as a battery of 1.5 volt.
The voltage can push the electrons into a component, like a resistor, creating a current. Usually, the voltage and the current are related by a formula (see impedance), Voltage is directly proportional to the current. If the voltage increases, the current also increases by the same amount.
Note that there must be both voltage and current to get power (energy). For example, a wire can have a high voltage on it, but unless it's connected, nothing will happen (that's why birds can land on high voltage lines such as 12kV and 16kV without problems).
There are two types of voltage, DC voltage and AC voltage. The DC voltage (Direct Current voltage) always has the same polarity (positive or negative), such as in a battery. The AC voltage (Alternating Current voltage) alternates between positive and negative. For example, the voltage from the wall socket changes polarity 60 times per second (in America). The DC is typically used for electronics and the AC for motors.
Mathematical definition
Mathematically, the voltage is the amount of work needed to move a charge of 1 coulomb from one position to the other.
 [math]V = I R[/math]
Where V=Voltage, I=Current, R=Total resistance.
Ground voltage
Voltage is always measured between two points, and one of them is often called the "ground", or the zero volt (0V) point. In most AC electrical installation there is eventually a connection to the real ground through a water pipe or any convenient metallic conductor buried underground, using the whole planet Earth as a reference voltage. In DC circuits, the negative end of a battery is often called the "ground", even though there is no connection to the real ground. Lightning rods are normally "grounded" through a copper cable on the side of a building and connected to a spike planted into the ground.
Measuring tools
Some of the tools for measuring the voltage are the voltmeter and the oscilloscope.
The voltmeter measures the voltage between two points and can be set to the DC mode or the AC mode. The voltmeter can measure the DC voltage of a battery for example (typically 1.5V or 9V), or the AC voltage from the power socket on the wall (typically 120V).
For more complex signals, an oscilloscope can be used the measured the DC and/or AC voltage, for example to measure the voltage across a speaker.
Potential Difference
The voltage, or potential difference from point a to point b is the amount of energy in joules (as a result of electric field) required to move 1 coulomb of positive charge from point a to point b. A negative voltage between points a and b is one in which 1 coulomb of energy is required to move a negative charge from point a to b. If there is a uniform electric filed about a charged object, a negatively charged objects will be pulled towards higher voltages, and a positively charged object will be pulled towards lower voltages. The potential difference/Voltage between two points is independent of the path taken to get from point a to b. Thus, the voltage from a to b + the voltage from b to c will always equal the voltage from a to c.

