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Electric Field and Potential

Electric Fields

Electric fields exist in a space where a charge experiences an electrical force. This force may derive from another single charge, or a charged sphere or parallel plates.

Electric fields and electric forces are vectors, since they have direction. By convention, a force away from a positive charge is considered positive.

$$E↖{→} = F↖{→}/q$$

where F is the force acting on a charge q in an electric field E. The unit of the electric field is $N C^{-1}$ (newtons per coulomb).

Electric Field Lines

Spherical charge
Spherical charges have electrical fields like the spokes of a bicycle wheel

If the forces were measured at every point around a charged sphere, it would be possible to draw a diagram of concentric circles (iso-force lines). However, it is also useful to draw lines which indicate the directions of the vectors of force between two or more charged bodies. These vector lines map out the field lines of the electric field.

If a metallic sphere is negatively charged, the lines of force point towards the sphere, indicating that a positive charge released near the sphere would be attracted to the sphere.

With a positively charged sphere, the field lines point away, indicating that a positive charge would be repelled.

Negative charges move in the opposite direction to the field lines, since they are attracted by positive charges.

The field inside a charged conductor is zero, because the sum of all the charges across the sphere equals zero everywhere within the sphere.

A point charge is like a very small sphere. It makes it easier to understand fields if we imagine the force starting from a single point, rather than spread across a large surface.

Just like a sphere, a single point charge will have electric field lines coming out from it like the spokes of a bicycle wheel.

Point charges
Point charges: lines of force indicate which direction a positive charge would move

Electric Force

It is useful to describe an electric field in terms of the strength of its force on a unit charge. This is:

$$F = k⋅{Qq}/{r^2}$$

where Q is the charge on the electric field source, q the charge on which the force is being measured, r is the separation distance between Q and q, and k is Coulomb's constant, which is defined as $1/{4πε_0} = 8.988 × 10^9 N m^2 C^{-2}$, where $ε_0$ is the permittivity of free space: $ε_0 = 8.85 ⋅ 10^{-12} N^{-1} m^{-2} C^{2}$

Since the electric field is the force per charge, $E↖{→} = F↖{→}/q$, we can state that:

$$E_p = k⋅Q/{r^2}$$

where $E_p$ is the magnitude of the electric field of a point or spherical charge Q at distance r.

Parallel plates
Plates with opposite charges have an electric field between them that never changes intensity

Two oppositely charged parallel plates have an electric field between them that is constant:

$$E = V/d$$

where E is the magnitude of an electric field between two parallel plates with potential difference (voltage) V, and separation d.

$$W = q⋅ΔV$$

where W is the work done when moving a charge q through a potential difference of ΔV.

$$U = qV$$

where U is the electric potential energy of a charge at a point where there is a potential V.

Electric fields and electric potential are expressed as:

$$E = {ΔV}/{Δr}$$

The Maths

The strength of an electric field is $E = F/q$, where F is the force (newtons N) and q is the charge (coulombs C). The unit of field strength is therefore N/C.

One proton has a charge of $1.6 × 10^{-19}$ coulomb (C), and one electron has a charge of $1.6 × 10^{-19}$ coulomb (C).

Flux of an Electric Field

Flux is a measure of the number of field lines through a given surface area. For example, a rotating coil in a magnetic field will have a flux which varies from maximum (at 90° to the field lines) to zero (when the two sides of the coil are both in the field line plane).

There is flux with both an electrical field and a magnetic field:

The definition of flux for the electric field E for a Gaussian surface with area A is:

$$Φ = ∑ E ⋅ ΔA$$

Law of Conservation of Energy

A charge moving in an electric potential satisfies the law of conservation of energy:

$$1/2mv_A^2 + qV_A = 1/2mv_B^2 + qV_B$$

where $1/2mv_A^2$ is the kinetic energy of the particle prior to the move from A to B in the electric field, and $qV_B - qV_A$ is the work done on the particle with charge q.

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