Institutt for fysikk, NTNU TFY4155/FY1003: Elektrisitet og magnetisme Spring 2005

Summary, week 4 (January 25 and 26)

Electric field lines [FGT 22.2; YF 21.6; TM 21.5; AF 21.6; LHL 19.6; DJG 2.2.1]

• provides a visual image of E in a given region of space

• E is everywhere tangential to the field lines

• the electric field strength (i.e. |E|) is everywhere proportional to the density of field lines, i.e., number of field lines pr unit area

Consequences of this are, for instance:

• the field lines are directed radially away from positive (point-) charges and radially towards negative charges

• the same number of field lines start on a charge +Q and end on a charge −Q

Electric dipole [FGT 22.1; YF 21.7; TM 21.4; AF 21.11; LHL 19.10; DJG 2.2.1, 3.4.2]

If we have two charges q and −q separated by some distance, we have an electric dipole. The (distance) vector d from the negative charge −q to the positive charge q describes how the two charges are located with respect to each other. The dipole moment, p, of the dipole is then defined like this:

p = qd

Thus, the dipole moment is a vector pointing from the negative towards the positive charge, with magnitude equal to the product between the value of the charge q and the distance d. Unit for electric dipole moment: [p] = [qd] = Cm.

Electric potential [FGT 24.2; YF 23.2; TM 23.1; AF 21.9; LHL 19.9; DJG 2.3.1, 2.3.2, 2.4.1]

We have a conservative force F if the work ∫_{A}^{B} F ·dl is independent of the path followed between the starting position A and the end position B. Examples of conservative forces are: The gravitational force between two masses. The electrostatic force between two charges. Example of non-conservative force: Friction.

More generally, we have a conservative vector field G provided that the line integral ∫_{A}^{B} G · dl is independent of the integration path between A and B.

For a conservative vector field G, we always have:

∮ G · dl = 0

where ∮ denotes integral around a closed path in space.

For a conservative force F we have a potential energy U so that the work done by F on ”the system” (e.g. the charge being moved) by a displacement from A to B corresponds to the change in the potential energy of the system:

ΔU = U_{B} − U_{A} = − ∫_{A}^{B} F · dl

[Check of the sign: A displacement of a mass m upwards, i.e., against the gravitational force mg, results in an increase in the potential energy. At the same time, we have F · dl < 0, which means that the sign is OK!]

Just as it was convenient to introduce the electric field E = F /q = electric force pr unit charge, it is now convenient to introduce electric potential as potential energy pr unit charge:

V = U/q

Unit for electric potential: [V] = [U/q] = J/C ≡ V (volt)

The relation between electric potential V and the electric field E [FGT 24.2; YF 23.2; TM 23.1; AF 21.10; LHL 19.9; DJG 2.3.1]

A charge q that is influenced by an electrostatic force F will have a difference in its potential energy

ΔU = U_{B} − U_{A} = − ∫_{A}^{B} F · dl

between the two points A and B. Then the difference in electric potential ΔV between the points A and B must be

ΔV = V_{B} − V_{A} = \frac{ΔU}{q} = − ∫_{A}^{B} \frac{F · dl}{q} = − ∫_{A}^{B} E · dl

An alternative unit for electric field: [E] = [V/l] = V/m

Note that while electric field is a vector, the electric potential is a scalar.

Note: Only differences in electric potential (and in potential energy) have physical meaning. We are free to choose where we want to have V = 0. A common choice is: V (r → ∞) = 0. Then, for the potential in a point P:

V_{P} = − ∫_{∞}^{P} E · dl

(We cannot always choose V (r → ∞) = 0. We will see some examples of that later.)