AHL Electric fields

In an electrical circuit, electrons move due to potential difference. Electric fields caused by charged plates, point charges or charged spheres cause lone charges to accelerate. This changes their energy.

Key Concepts

Uniform fields

Field lines

A uniform field has a constant field strength. Field lines are equidistant and parallel, and show the direction of force acting on a positive test charge. Uniform electric fields are formed between oppositely charged parallel plates.

Beyond the edges of the plates, the field would no longer be uniform, but these are not drawn.

Field strength

Electric field strength is defined as the force per unit charge:

\(E={F_e\over q}\)

Field strength is equal to the magnitude of the potential gradient:

\(E={V_e\over d}\)

These equations are often combined, for example to find the force acting on a given charge:

\({F_e\over q}={V_e\over d}\)

\(E\) is electric field strength (NC^-1 or Vm^-1)
\(F_e\) is electric force (N)
\(q\) is the charge placed in the field (C)
\(V_e\) is the potential difference between the plates (V)
\(d\) is the distance between the plates (m)

Calculating energy changes

The energy changes that take place in electric fields are between electric potential energy and kinetic energy:

Work can be done by a charge on the field if it moves towards the plate of like charge; the charge loses kinetic energy and gains potential energy.
Work is done by the field on the charge if it is attracted to the plate of opposite charge; the charge loses potential energy and gains kinetic energy.

Since electric field strength is constant, the force acting on the charge is constant everywhere. Work is done when a charge moves along a field line, perpendicular to the plates:

\(W=F_{e}s=\Delta E_p=q\Delta V_e\)

\(W\) is work done (J)
\(F_e\) is electric force (N)
\(s\) is displacement parallel to field lines (m)
\(\Delta E_p\) is change in potential energy (J)
\(q\) is charge (C)
\(\Delta V_e\) is potential difference

No work is done parallel to the plates. This is because the charge would be moving along an equipotential surface.

Equipotential surfaces

Equipotential surfaces are planes of constant potential. They are always perpendicular to the field lines. For a uniform field, equipotentials are equidistant and parallel, here shown in red:

Essentials

Radial fields

Field lines

Radial fields are produced by point charges or charged spheres. Field strength increases with proximity to the charge. This is indicated by closer field lines. The direction of the field lines shows the direction of force acting on a positive test charge placed in the field.

Field strength

As radial fields are non-unifom, electric force and field strength are not constant. Electric force is found using Coulomb's law:

\(F_e=k{Qq\over r^2}\)

Electric field strength is the force per unit charge:

\(E={F_e\over q}=k{Q\over r^2}\)

\(E\) is electric field strength (NC^-1)
\(F_e\) is electric force (N)
\(q\) is the charge placed in the field (C)
\(k\) is a constant (8.99×10⁹ N m² C⁻²)
\(Q\) is the charge producing the field (C)
\(r\) is the distance between the centres of the charges (m)

Calculating energy changes

Electric potential is the work done per unit charge in bringing a positive test charge from infinity to a point in an electric field at a small constant speed.

As force is not constant, electric potential is found by integrating the force with respect to distance from infinity:

\(V_e={\int_\infty^r F_e\,\mathrm{d}r\over q}= {\int_\infty^r {kQq\over r^2}\,\mathrm{d}r\over q}\)

\(\Rightarrow V_e=k{Q\over r}\)

\(V_e\) is electric potential (JC^-1 or V)
\(k\) is a constant (8.99×10⁹ N m² C⁻²)
\(Q\) is the charge producing the field (C)
\(r\) is the distance between the centres of the charges (m)

As with energy and work done, potential is a scalar quantity.

Energy is changed when an object is moved through a distance by a force. When a charge is moved into or out of a non-uniform electric field, it will be attracted or repelled. Therefore, work is done and energy becomes stored in the charge itself. This energy is electrical potential energy and is measured in J.

\(E_p=qV_e=k{Qq\over r}\)

Equipotential surfaces

The equipotential surfaces in a radial field are spherical, shown as circles on a 2-dimensional diagram. As potential varies with \(1\over r\) from the centre of the charge, the equipotentials increase in spacing outward from the centre.

Work is done when a charge is moved between equipotentials:

\(W=q\Delta V_e=q\times kQ({1\over r_f}-{1\over r_i})\)

\(W\) is work done (J)
\(q\) is the moving charge (C)
\(\Delta V_e\) is electric potential difference (JC^-1 or V)
\(k\) is a constant (8.99×10⁹ N m² C⁻²)
\(Q\) is the charge producing the field (C)
\(r\) is the initial or final distance between the centres of the charges (m)

Potential gradient is the ratio of the difference in potential to the distance between the equipotential surfaces. Its magnitude is equal to electric field strength:

\(E=-{\Delta V_e\over \Delta r}\)

\(E\) is electric field strength (NC^-1)
\(\Delta V_e\) is potential difference (JC^-1 or V)
\(\Delta r\) is the distance between the potentials (m)

Inside a charged sphere

So far we have examined only the electric field outside of the charge producing the field. What about the field strengths and potentials at radii less than the radius of the sphere?

A sphere may have uniform distribution of charge:

By Ag2gaeh - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=49540033

The charge enclosed within the sphere by a radius \(r\) in comparison with the total radius \(R\) is related to the volume:

\({q\over Q}={r^3\over R^3}\)

\(E=k{q\over r^2}=k{Qr\over R^3}\)

\(V_e=k{q\over r}=k{Qr^2\over R^3}\)

The sphere may alternatively be a conductor. In this case the charge would move to the surface. In this situation, the field at any radius within the sphere would be zero; the potential would be the same as at the surface.

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