Electric Field and Dipoles: Quick Revision

A brief note on Electric Field and Dipoles. Read it before a test. Try each question at the end before you open its answer.

The field and field lines

• Definition: $E = \dfrac{F}{q_0}$, in N/C, for a small positive test charge. The force on a charge $q$ is $F = qE$.
• A point charge: $E = \dfrac{kQ}{r^2}$, with $k = 9.0 \times 10^{9}\ \text{N m}^2/\text{C}^2$. It points away from $+Q$ and towards $-Q$.
• Superposition: the field of several charges is the vector sum of the fields of each one.
• A charge in a field: $a = \dfrac{qE}{m}$. A negative charge goes against the field.
• Field lines: they start on positive charges and end on negative ones, never cross, and never form closed loops. Close lines mean a strong field.
• A zero of the field: between two like charges, and outside two unlike charges on the side of the smaller. Use $\dfrac{x}{d - x} = \sqrt{Q_1/Q_2}$.

Fields of charge distributions

• Densities: $\lambda = Q/L$, $\sigma = Q/A$, $\rho = Q/V$.
• Ring, on the axis: $E = \dfrac{kQz}{(R^2 + z^2)^{3/2}}$. It is zero at the centre and greatest at $z = R/\sqrt{2}$.
• Rod, on the perpendicular bisector: $E = \dfrac{kQ}{r\sqrt{r^2 + (L/2)^2}}$. For a very long wire, $E = \dfrac{2k\lambda}{r}$.
• Disc, on the axis: $E = \dfrac{\sigma}{2\varepsilon_0}\left(1 - \dfrac{z}{\sqrt{z^2 + R^2}}\right)$. Near a large sheet, $E = \dfrac{\sigma}{2\varepsilon_0}$.
• Symmetry: the parts of the field across the axis cancel, and the parts along it add.

Dipoles

• Moment: $p = q \times 2a$, in C m, from $-q$ to $+q$.
• On the axis, far away: $E = \dfrac{2kp}{r^3}$, along $\vec{p}$.
• On the equatorial line, far away: $E = \dfrac{kp}{r^3}$, against $\vec{p}$.
• In a uniform field: net force zero, torque $\tau = pE\sin\theta$, energy $U = -pE\cos\theta$. Stable at $\theta = 0^\circ$, unstable at $180^\circ$.
• Small swings: $T = 2\pi\sqrt{\dfrac{I}{pE}}$. In a non-uniform field there is also a net force, towards the stronger field.

A worked example

A rod $0.40$ m long carries $8.0$ nC spread evenly. Find the field $0.30$ m from it, on the line at right angles through its middle.

1. The square root. $\sqrt{r^2 + (L/2)^2} = \sqrt{0.090 + 0.040}\ \text{m} = 0.361\ \text{m}$.
2. The field. $E = \dfrac{kQ}{r\sqrt{r^2 + (L/2)^2}} = \dfrac{9.0 \times 10^{9} \times 8.0 \times 10^{-9}}{0.30 \times 0.361}\ \text{N/C} = 665\ \text{N/C}$, away from the rod.

Practise with new numbers

Check yourself

1. The field of a point charge is $90$ N/C at a distance of $2.0$ m. What is it at $6.0$ m?

2. A small dipole has a field of $12$ N/C at a distance $r$ on its equatorial line. What is the field at the same distance on its axis?

3. A dipole is at $90^\circ$ to a uniform field. What is the torque, and what is the net force?