How is voltage related to electric fields?

Electric Field (E):

An electric field is a vector field that exists around electric charges. When a positive charge is placed in space, it creates an electric field that points away from it. Similarly, when a negative charge is placed, the electric field points toward it. The electric field strength (E) at any point in space is defined as the force (F) experienced by a positive test charge (q) placed at that point, divided by the magnitude of the test charge:

Electric Field (E) = Force (F) / Test Charge (q)

The unit of electric field is volts per meter (V/m).

Voltage (V):

Voltage, also known as electric potential difference, is a measure of the energy required to move an electric charge between two points in an electric field. It is often referred to as the "electrical pressure" that drives charges in a circuit. Voltage is measured in volts (V).

The voltage (V) between two points A and B in an electric field is equal to the work done (W) in moving a charge (q) from point A to point B, divided by the magnitude of the charge:

Voltage (V) = Work done (W) / Charge (q)

The relationship between voltage and electric field is described by the integral form of Gauss's law for electrostatics:

∮E · dA = (1/ε₀) * Q

where:

∮E · dA represents the closed surface integral of the electric field over an enclosed surface (A).

ε₀ (epsilon naught) is the vacuum permittivity, a fundamental constant in electromagnetism.

Q is the total charge enclosed by the surface.

In simpler terms, the integral of the electric field over a closed surface is equal to the total charge enclosed divided by the vacuum permittivity. This equation shows how the distribution of charges creates an electric field in space.

In summary, voltage and electric fields are related through the concept of work done in moving charges in an electric field. The electric field creates a potential difference (voltage) that causes charges to move and establish current flow in electric circuits.