OCW/Electricity and Magnetism/Lecture 2

Electric Field

Definition

We have a charge Q at a fixed location, and a different charge q called the test charge at location P. The charges are separated a distance r, and we have a unit vector $\hat r$ that points from Q to q. This charges will experiment some force according to Coulomb's Law. The force experimented by q is:

$\vec F = \frac{qQk}{r^2} \hat r$

The Electic Field E at the point where the test charge q is set, is:

$\vec E = \frac{\vec E}{q} = \frac{Qk}{r^2} \hat r$

The Electric Field is also a vector, so can be represented with an arrow. By convention, and following Coulomb's Law, the test charge is positive, so if the charge Q is also positive, the electric field is away from Q, and if Q is negative, the field is towards Q.

The unit of E is Newtons divided by Coulombs.

Graphical representation

Electric field is a vector, so you can use representations using arrows. By convention, the arrows are poiting away from a (positive) charge, because it's the direction that a positive test charge will experiment a force in.

Superposition

In case that there are several charges Qi, you can calculate the separate effect of the electric field of each charge in a point P, and then sum the resultant vectors. This is the superposition principle.

Once I know the field at a point, I know that if I put a charge q there, the force that acts upon it is $\vec F = q \vec E$.

Example: +3 and -1 charges

See the field configuration when you have a +3 and a -1 charges. When you are far away, the lines are pointing out, as if only there was a +2 charge.

Field Lines

Field lines indicate the direction of the force that a field will create on a positive test charge (if it's negative, the force will flip over). The force will be tangential to the lines. If the lines come closer and closer, the field becomes stronger. If the lines spread out, the field becomes weaker.

Remember that field lines are not trajectories. If the lines are straight and parallel, they can match, but is not the normal case.

Another example: dipoles

A dipole is a configuration in which there is both a positive and a negative charge. When you are far away, the force fades faster than with only a charge (r3 instead of r2), because one charge counteracts the other.

Experiment: a dipole with two metallic spheres

Create a dipole with two spheres.

1. Approximate a charged rubber rod to two conducting connected spheres.
2. The electrons in the spheres try to go away from the rod, so more electrons go the sphere that is further away from the rod.
3. When you separate the spheres, one sphere has more electrons, and the other has less, so one becomes positively charged, and the other negatively charged.

Experiment: dipole rotation

Dipoles experiment a torque inside a field, because the positive charge experiments a force in one direction, and the negative in the oposite.

1. Create a dipole through induction.
2. Put the dipole in a field, so it becomes oriented in the direction of the field.

Experiment: field lines with grass seeds

Grass seeds are small dipoles, so if you put them in oil, they can draw field lines very nicely.

Experiment: Moving charge

Starts at 45:45. Funny experiment that charges a balloon and shows how it permanently moves from one point to another.