# Potential difference

In the physical sciences, potential difference is the difference in potential between two points in a conservative vector field. In engineering, it is sometimes described as the across variable, where flux is the through variable. Within the context of electricity, the conservative vector field is the electrostatic field. Between any two points in the field, there is defined an electric potential difference where the unit is a volt. This electric potential difference is referred to as the voltage between the two points.

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## Production

The product of the flux and the potential difference is the power, which is the rate of change of the conserved quantity, e.g., energy. In fluid systems the potential difference is the difference in pressure. In thermal systems the potential difference is the difference in temperature. In mechanics, the potential difference is the difference in gravitational potential between two points. In electrical engineering the potential difference is the voltage, i.e. the difference between the initial and final points of an electrostatic potential.

### Electrical definitions

A potential difference between two points gives rise to a "force" called an electromotive force or emf that tends to push electrons or other charge-carriers from one point to the other. A potential difference is generated between the ends of an electrical conductor that moves perpendicular to a magnetic field. Between two points in an electrical circuit the potential difference is equal to the difference in their electrical potentials.

The potential difference is defined as the amount of work per charge needed to move electric charge from the second point to the first, or equivalently, the amount of work that unit charge flowing from the first point to the second can perform. In the SI system of units, potential difference, electrical potential and electromotive force are measured in volts, leading to the commonly used term voltage and the symbol V. Named after Alessandro Volta, one volt is defined to be one joule of energy per coulomb of charge.

The potential difference between two points a and b is the line integral of the electric field E:

[itex]V_a - V_b = \int _a ^b \mathbf{E}\cdot d\mathbf{l} = \int _a ^b E \cos \phi dl.[itex]

If one thinks of an electrical circuit in analogy to water circulating in a network of pipes, driven by pumps in the absence of gravity, then the potential difference corresponds to the pressure difference between two points. If there is a pressure difference between two points, then water flowing from the first point to the second will be able to do work, such as driving a turbine.

Voltage is additive in the following sense: the voltage between A and C is the sum of the voltage between A and B and the voltage between B and C. Two points in an electric circuit which are connected by an (ideal) conductor without resistance have a potential difference of zero. But other pairs of points may also have a potential difference of zero. If two such points are connected with a conductor, no current will flow through the connection. The various voltages in a circuit can be computed using Kirchhoff's circuit laws.

### Instruments

Common sources of emf are the battery, the electrical generator and the capacitor. Instruments for measuring potential differences include the voltmeter, the potentiometer (measurement device), and the oscilloscope. The voltmeter works by measuring the current through a fixed resistor, which, according to Ohm's Law, is proportional to the potential difference across it. The potentiometer works by balancing the unknown voltage against a known voltage in a bridge circuit. The cathode-ray oscilloscope works by amplifying the potential difference and using it to deflect an electron beam from a straight path, so that the deflection of the beam is proportional to the potential difference.

## History

In 1800, as the result of a professional disagreement over the galvanic response advocated by Luigi Galvani, Alessandro Volta developed the so-called voltaic pile, a forerunner of the battery, which produced a steady electric current. Volta had determined that the most effective pair of dissimilar metals to produce electricity was zinc and silver. In the 1880s, the International Electrical Congress, now the International Electrotechnical Commission (IEC), approved the volt for electromotive force. The volt was defined as the potential difference across a conductor when a current of one ampere dissipates one watt of power.

## Analogies

Voltage is analogous to the hydrostatic pressure of a fluid in a pipe. It also explains the "dimensions" of voltage. Consider the potential energy of compressed air being pumped into tank. The energy increases with each new increment of air. Pressure is that energy divided by the volume, which we can understand intuitively. Now consider the energy of electric charge (measured in coulombs) being forced into a capacitor. Voltage is that energy per charge, so voltage is analogous to a pressure-like sort of forcefulness. Also, dimensional analysis tells us that voltage ("energy per charge") is charge per distance, the distance being between the plates of the capacitor. (See also Hydraulic analogy)
Reference: page 16 of "Industrial Electronics," by D. J. Shanefield, Noyes Publications, Boston, 2001.

Another analogy can be made with energy by itself (or work). You need 1 joule of energy to produce 1 joule of work. (A joule of work is produced by constantly applying 1 newton of force to move something 1 meter away). If we removed "C" in "1 V = 1 J/C" leading only to "1 J", we would think of voltage as simply energy ready to produce some work (moving a lot of electric charges). But when we talk about voltage, we don't need to know how many coulombs will actually move at the end, we just need to push all of them equally (as many as they are) while the device is on; we don't need the absolute value of energy, but the energy needed to move each electron (each coulomb). Thus, we normalize it by dividing over "C". If we removed some electrons without removing any of that energy, the voltage would be greater.

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