Charge and Field

An electric field is a region in space where a charged particle experiences an electric force. Electric field lines show the direction that a small positive test charge would experience a force if it were placed in the field. The greater the density of field lines, the greater the force.

Force and Field Strength

The force due to the electric field, $F$ is given by:

$$F=\frac{kQq}{r^2}\text{where}k=\frac{1}{4\pi {\epsilon }_0}$$

The force has an inverse-square relationship with distance. The electric force is different from gravity in that it can be either attractive or repulsive depending on the charges of the particles.

The field strength, $E$, is the force per unit charge, and is given by:

$$E=\frac{F}{q}=\frac{kQ}{r^2}$$

In a uniform electric field, the field strength throughout the field is constant, similar to how the field strength in a uniform gravitational field is constant. The force on an object in a uniform field is also constant. In a uniform field:

$$E=\frac{-\text{d}V}{\text{d}r}=\frac{V}{d}$$

The electric force and field are both vector quantities, whereas energy and potential are both scalar quantities.

Energy and Potential

The electric potential energy, $U$, is given by the negative integral of force with respect to distance:

$$U=\frac{kQq}{r}$$

Again, this is similar to $U=\frac{GMm}{r}$ for gravitational potential energy. The potential energy has an inverse proportional relationship with distance.

The electric potential, $V$, is the energy per unit charge, and is given by:

$$V=\frac{U}{q}=\frac{kQ}{r}$$

Electric potential $V$ and field strength $E$ are both properties of the field itself (as they are per unit charge), whereas potential energy $U$ and force $F$ act on a specific object within the field.

Graphically, the electric potential energy is the area under a force-distance graph between two points (by $W=Fs$) (possibly requiring a negative sign). Thus, the force is related to the gradient of a energy-distance graph.

Similarly, the potential is the area under a field strength against distance graph, and the field-strength is related to the gradient of a potential-distance graph.

Equipotential surfaces

Equipotential surfaces are surfaces along which all points have the same potential. They are always perpendicular to the direction of field lines. In a uniform field, equipotential surfaces are equally spaced.

Force on a moving charge

Consider a particle moving with charge $q$ moving through a magnetic field $B$ at velocity $v$. The ‘current’ produced by this particle in a time $\Delta t$ is $\frac{\Delta q}{\delta T}$, and substituting into $F=BIL$ gives:

$$F=BIL=B\left(\frac{\Delta q}{\Delta t}\right)L=Bq\left(\frac{\Delta L}{\Delta t}\right)=Bqv$$

For a charged particle moving in a magnetic field, $F=Bqv$. The left-hand rule can be used to find the direction of the force (taking care that current in the left-hand rule is conventional current, i.e. the direction of flow of positive charge).

Equating this force with the centripetal force in circular motion, $F=\frac{m{v}^2}{r}$, can be used to find the radius of circular motion for a charged particle moving in a magnetic field.

Electron charge

The charge on the electron is discrete; all charged particles have charge equal to some integer multiple of the elementary charge^¹, $1.60\times 10^{-19}\text{C}$.

This was first shown in Millikan’s oil drop experiment. In the experiment:

• An atomiser creates a mist with small droplets of oil.
• The droplets are ionised by X-rays, causing the droplets to become charged.
• A potential difference is applied across two metal plates, creating an electric field.
• The oil droplets are observed as they fall, and balancing forces allows for the calculation of the charge on the oil droplets.

Footnotes

1. Footnote on elementary charge Other than quarks (which don’t exist by themselves). ↩