Point Charges
[drill and practice]

An electric field is the region surrounding a charged particle, Q, where another charged particle with experience either a force of attraction or repulsion.  For point charges, the electric field lines are radial, getting ever farther apart as you get farther from the point charge itself.  These fields are NOT uniform, but are examples of inverse square fields

E = kQ / r ².

Dimensional analysis reveals that the units on E, the electric field strength, should be

Intuitively, the electric field strength measures the amount of force, in newtons, experienced by a coulomb of charge when it is placed at a particular position within an electric field.

Electric field lines point in the direction in which a positive test charge would respond to the electrostatic force; that is, away from positive charges and towards negative charges.  In the following diagram, Q is positive, since the field lines are pointing away from Q.  If Q had been negative, then the field lines would have pointed towards Q.  Two size of two charges can be compared by noting the relative number of field lines surrounding each one.  If a second charge with only 10 fields lines was compared to the diagram provided below, then it would indicate that the second charge was only ½ as large, since 10 is half of 20.

Fields between oppositely charged particles are attractive and are elliptical in shape; while fields between similarly charged particles are repulsive and hyperbolic in shape.

oppositely charged particles left is positive; right is negative	similarly charged particles both are positive

A convenient way to remember the properties of an electric field are to use analogies to gravitational fields.  A gravitational field is the region surrounding a massive object in which another object with mass will experience a force of gravitational attraction.  One important distinction between electrical fields and gravitational fields is that electrical fields can be both attractive and repulsive; whereas gravitational fields are only attractive.

In the chart above, you can see that both fields are inverse square relationships.  That the electrical field strength, E, has the same configuration as the gravitational field strength, E.  That in each case, the force experienced by a second object equals the product of either that object's mass times the gravitational field strength or that object's charge times the electrical field strength.  The direction of the gravitational field is defined as the direction a second object with mass would be attracted; whereas the direction of an electrical field is defined as the direction a positive test charge would respond.

When charged particles are moved from one position in an electric field to another position, a new unit of measurement is needed.  A volt represents the amount of work per unit charge required to move a charge between two positions in an electric field. If it takes 1 joule of work to move 1 coulomb of charge between positions A and B in an electric field, then positions A and B have a potential difference of 1 volt.  Voltage is a scalar property of an electric field, it has no direction, only magnitude.

In general, 1 volt = 1 joule / 1 coulomb
W = qDV

For a point charge the absolute potential of any position in its electric field is calculated using the equation

V_abs = k Q / r.

When the charge creating the field is positive, the voltage is positive; when the central charge is negative, the voltage is negative.

For example: If the central charge was 10 mC, what is the potential at surface A where
r_A = 3 meters and at surface B where r_B = 1 meter?

V_A = k Q / r = ( 9 x 10⁹ )(10 x 10^{- 6} ) / 3 = 3 x 10⁴ volts
V_B = k Q / r = ( 9 x 10⁹ )(10 x 10^{- 6} ) / 1 = 9 x 10⁴ volts
Since B is closer to the charge it is at a higher potential than surface A.

Surfaces which connect points that are at the same absolute potential, or voltage, are called equipotential surfaces.  In the diagram of the point charge shown above, two equipotential surfaces have been labeled, A and B.  All equipotential surfaces meet field lines at right angles.  The closer together two equipotential surfaces are to each other, the more rapid the change in voltage; this signifies a stronger electric field.

Note that the electric field strength, E, can be measured in either the units V / m, or equivalently, N / C.

Charge configurations to check for whether or not net E and/or net V equal zero in the center of a square.

A	B	C	D

If a 2 nC charge where to be brought in from infinity and placed on surface A, the amount of work done on the 2 nC charge would equal

We say that the 2 nC charge has gained an electric potential energy of EPE_A = 6 x 10^-5 J. 

Similarly, the amount of work done on the 2 nC charge to bring it in from infinity and place it on surface B would equal

We say that the 2 nC charge has gained an electric potential energy of EPE_B = 1.8 x 10^-4 J. 

The difference between the 2 nC's electric potential energy at B and its electric potential energy at A would represent the work required to move the 2 nC change from a position on surface A to a position on surface B.

W = q D(V_abs ) = DEPE

In essence, positive voltages mean that an external agent must do work to move a positive charge to a new position in the field.  Negative voltages mean that the field would be doing all of the work to move a positive charge to its new position in the field.  By definition, the absolute potential at a point infinitely far from a point charge is defined to be zero. If a system contains more than one charge, then the EPE of the system is the sum or the EPE of each charge.

By definition, charges flow from points of high potential to points of low potential. That is, when free to move, a positive charge would instinctively flow from surface B (high potential) to surface A (lower potential). Because of this, work done by electric fields (that is, when the charge moves along a field line in the direction of the field) results in a charge LOSING electric potential energy - that is, the electrostatic force causes a charge to move to positions of lower potential and less electrical potential energy consequently gaining KE.

An analogy can be formed between equipotential surfaces and altitudes from the surface of the Earth. At any given altitude, an object with mass has a certain amount of gravitational potential energy, PEg = mgh where h is measured from some arbitrarily set zero level (usually the base of the hill). An external agent must do work against the gravitational field whenever the object's height, altitude, is increased  - this results in the object gaining PEg. The gravitational field does work on the mass whenever the object's height, altitude, is decreased resulting in the object losing PEg.

By comparing the aerial view with the side view, you can tell that when the surfaces are closer together on the left, it signifies that the altitude is changing more rapidly, that is, that the slope of the hill is steeper. But regardless of which side of the hill a person climbs, he will do the same amount of work and gain the same amount of PEg = mgh. Remember that this is the definition of a conservative field.

If instead, the aerial view where to be considered to be a series of equipotential surfaces, then the electric field is stronger on the left than on the right. Since the same changes in voltage occur in a smaller distance on the left side than on the right side. However, regardless of which direction a charge is moved from one surface to another, the same amount of work is done, since the charge gains or loses the same amount of electrical potential energy,

DEPE = q ( DV_abs )

The fact that these changes are path independent signifies that an electric field is also a conservative field -- that is, the only thing that counts is a comparison of the ending position to the initial position, not the path taken between the two points. By definition, the absolute potential at a point infinitely far from a point charge is defined to be zero. If a system contains more than one charge, then the EPE of the system is the sum or the EPE of each charge.

Use the following electric field map to discern if you understand the concepts of an electric field and electric potentials discussed so far.