Guide to electric field calculations

  1. The basic approach is that, at every position $\vec{r}$, we sum the contribution to the electric field from every charge.

    • This follows straight from the principle of superposition
    • If the charge distribution is continuous, we have to replace the sum over charges by an integral summing over infinitessimal charge elements
  2. Always start by drawing a clear diagram.
    • Show the coordinate axes
    • Clearly identify the charges
    • Show the point at which we are calculating the electric field $\vec{E}$
  3. Look for symmetries
    • Symmetries can make forces cancel, or dictate which direction the force must point in, as you will see in examples shortly
    • This can save a lot of work

Once you have calculated an electric field at a position $\vec{r}$, calculating the force on a charge $q$ in the same position is straightforward because $\vec{F} = q \vec{E}$.

Worked examples

In the lecture we will work through these examples.

  1. Two charges of $10\,{\rm nC}$ are placed on the $y$-axis with $y=+10\,{\rm cm}$ and $y=-10\,{\rm cm}$.

    a. Calculate the electric field at $x=20\,{\rm cm}$, $y=0\,{\rm cm}$.

    b. What is the force felt by a $1\,{\rm nC}$ charge at this location?

  2. A charge $Q$ is distributed uniformly around a narrow ring of radius $a$ in the $y$ – $z$ plane.

    a. Find the electric field at a point $P$ on the ring’s axis ($y=z=0$) at a distance $x$ from its centre.

    b. What is the approximate form of the $E$ field at a distance $x \gg a$ from the ring?

Note that the second question is an example of a charge distribution, where there are no individual point charges but rather a continuous distribution of charge.

Obviously, in reality, there are point charges at the atomic level, but we can often treat them as continuous distributions when we are looking at the field on larger scales.

Dipoles

A dipole is a special arrangement where two equal and opposite charges $+q$ and $-q$ are separated by a vector distance $\vec{d}$. By convention the vector $\vec{d}$ points towards the positive charge (i.e. away from the negative charge).

  • if we look on very large scales compared with $\vert \vec{d} \vert$, the charges look like they are basically on top of each other and then the principle of superposition says there is no electric field.
  • if we look on very small scales compared with $\vert \vec{d} \vert$, we can just think of the two charges as separate.
  • interesting things happen on the in-between scale, i.e. on scales comparable to $\vert\vec{d} \vert$.

Try dragging around these two equal and opposite charges, and see how the $\vec{E}$ field responds. Look at the field when the charges are far apart, close together, and even on top of each other. Can you explain what happens in each case? You can use the checkbox at the bottom left to switch between seeing the fieldline and ‘quiver’ visualizations.

The dipole moment describes the strength and direction of the dipole:

\[\vec{p} = q \vec{d}\]

Generally, when we talk about dipoles, we mean something where the distance between the charges $\vert \vec{d} \vert$ is fixed, as though there is a rigid pole connecting the two charges. The direction $\hat{\vec{d}}$, however, can change, i.e. the dipole can spin around.

Anything which is overall neutral but where the charge distribution is slightly asymmetric behaves a bit like a dipole – even if it doesn’t literally consist of two point charges.

For example:

  • The particular way the electrons orbit within a water molecule means that the side with two hydrogens is slightly positive, while the side with the oxygen is slightly negative.
  • So water molecules behave like electric dipoles.
  • Many molecules like salt (sodium chloride) dissolve in water because the positive end of water molecules surrounds the negative chloride ions, and the negative end surrounds the positive sodium ions.

As discussed previously charges don’t tend to separate very much. As a result dipoles are often the most important thing to consider when thinking about electric fields in materials. (More on this when we talk about materials next week.)

Forces on dipoles

Imagine you put a rigid dipole in a uniform electric field. What happens to it?

  • The net force acting on the dipole as a whole is zero.
  • But the force does not act in the same way across the dipole as a whole. The positive side of the dipole is pushed along the direction of the electric field; the negative side is pushed in the opposite direction.
  • When different forces apply to different parts of a rigid body we say there is a torque (refer back to your mechanics lectures, and/or to Sec 10.1 of Y&F).
  • Torque causes bodies to rotate.

This can actually be quite intuitive when you look at the direction that the different ends of the dipole are being pulled. Try playing with the example below to develop some intuition. The curved arrows show the direction and magnitude of the torque. You can untick the ‘rigid dipole’ option to get see the two separate charges and the forces on them.

By definition, torque $\vec{\tau}$ is

\[\vec{\tau} \equiv \vec{d} \times \vec{F}\]

where $\vec{d}$ is the distance to the origin around which the torque is being measured and $\vec{F}$ is the force being applied.

In this case, we take the origin to be the centre of mass of the dipole. There are actually two forces acting, one on either side of the dipole itself. We sum them to get the total torque. In the lecture we will see the resulting expression can be simplified to be

\[\vec{\tau} = \vec{p} \times \vec{E}\]

where $\vec{p}$ is the dipole moment.

Discussion point: In many ways, methane is a similar molecule to water, but its boiling point is much, much lower: $-162^\circ{\rm C}$. What might this have to do with water’s dipole moment (which methane lacks)? Could cold planets with liquid methane on them support life, playing the role that water plays here on Earth?