Supplementary Notes for PHYS 212: Spring 212

Electric Potential, Work, Electric Potential Energy:

Electric field of a point-like charge (a charge that doesn’t have structure, a proton or an electron is a good approximation, whereas a charged sphere is the opposite example which is not a point-like.) at a field point P which is located at a point r distance away from the point charge is

Note that there is nothing, no charge, at the field point.

This electric field can be assigned to any point around

any charge distribution (in this case Q is the charge

distribution).  Electric field’s direction

is along the line joining the field point to the source

of the field (i.e. Q in this case), and directed away

from the positive charge, and into the negative charge.

The picture depicts the positions and relative

magnitudes of the two electric fields.  E total

represents the addition of two electric field

vectors ( E+q and E-q).  Note that positive charges

field is smaller than that of the negative’s due

to different distances.

Another charge Q when placed at the

field point experiences a Coulomb force

From the +q and –q, which can be calculated as

This is a general formula that holds for any electric field (whereas kq/r2 holds for point charge only).  Also if Q is positive, F’s direction is the same as E’s.  If Q is positive, F’s direction is opposite to that of the E’s.

Note that, in the picture, the force on a positive

charge is opposite to that on the negative

charge even though both charges are placed

in the same electric field E.

Here PE is measured in Joules (J) and Potential V is measured in Volts (V).

• Work done on a charge by the electric field is path independent. This means that as long as you start at A end up at B, you can take any path between A and B, you will get the same value for work done.
• When a path is followed in the direction of the electric field lines, the electric potential drops (same as saying that electric field lines run from higher potential (positive terminal of the parallel plate capacitor) to lower potential (negative plate of the capacitor)).  For example in the picture provided below, VA > VB, therefore ΔV =VB-VA< 0 or negative.

When a charged particle follows the two paths (dotted lines above) depicted in the picture above, the work done by the field is the same for both of the paths.  In addition, if a negative charge, say an electron (charge –e) is carried from A to B, work done by the electric field is negative.  Because

•
When a charge is moved a distance d through a uniform electric field E, the work done on the charge by the field is

work along AC is zero since force is

perpendicular to path. Work along

CD is qEd, where d is the distance

CB.

Electric Potential of Point-like Charges:

Electric potential is a scalar quantity.  The electric potential of several charges at a given field point P is

Note that, the sign of the charge should

be included in all of your calculations that

involve PE, V or W.  The distance in

the above formula (r’s) are absolute values,

that is they are positive.

A negative charges electric potential will

be negative.  Potential doesn’t depend on

the direction. Potentials at points that are at equidistance from the charge all have the same potential value.  The union of points that have the same potential is called equipotential surface or line.  For example, for a point like charge, equipotential surfaces are concentric spheres centered on the charge.  A two dimensional projection gives equipotential lines.

Equipotential lines:

The picture depict the electric field

lines as well as equipotential lines

for a positive point like charge.  The

value of the potential for each line can be

calculated from the formula given above.

·        A charge can freely be moves on an equipotential line or surface.  This is because, on an equipotential line or surface ΔV = 0, therefore, W =-q ΔV = 0.  It means that the work done by the electric field when a charge moves on an equipotential line/surface is zero.

·        Electric field lines are perpendicular to the equipotential lines/surfaces.

·        A conductor’s surface is an equipotential surface.  This means that if a conductor is connected to a battery of 1.5 V at one point, all the points of the conductor at the potential of 1.5 V with respect to some reference point.