ELECTRIC POTENTIAL Spring, 2008

Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the electric force and accomplish the following tasks: Calculate V if we know the corresponding electric field Calculate the electric field if we know the corresponding potential V Determine the potential V generated by a point charge Determine the potential V generated by a continuous charge distribution Determine the electric potential energy U of a system of charges Define the notion of an equipotential surface Explore the geometric relationship between equipotential surfaces and electric field lines Explore the potential of a charged isolated conductor (24 - 1)

Picture a Region of space Where there is an Electric Field Imagine there is a particle of charge q at some location. Imagine there is a particle of charge q at some location. Imagine that the particle must be moved to another spot within the field. Imagine that the particle must be moved to another spot within the field. Work must be done in order to accomplish this. Work must be done in order to accomplish this.

What (or who) must do this work? An external agent (person) An external agent (person) The Field itself The Field itself Either of the above Either of the above

Electric Potential We will be dealing with  Work  Energy & Conservation Work must be done to move a charge in an electric field.  Let’s do a demo ….

I need some help.

What we will do …. For the moment, assume the charge has MASS. (It may not.) Assume the charge is initially stationary. The charge is to be moved to the left. The charge is to be moved at CONSTANT velocity. + charge E Mr. ExternalMrs. Fields

During this process, who is pushing? Mr. External Mrs. Fields

ENERGY is required to bring the charge up to speed (if it has mass). ENERGY is required to bring the particle back to rest (if it has mass). The sum of these two is ZERO.

During this process, who is actually doing work? Mr. External Mrs. Fields Both of them Neither of them.

Clearly Both are doing work. BOTH are applying a force through a distance. BOTH get tired!

About the work that they do.. Mrs. Fields does more work than Mr. External. Mr. External does more work than Mrs. Fields. Both do the same amount of work. Each does the negative amount of work than the other does.

So, when we move a charge in an Electric Field.. Move the charge at constant velocity so it is in mechanical equilibrium all the time. Ignore the acceleration at the beginning because you have to do the same amount of negative work to stop it when you get there.

Summary-- When an object is moved from one point to another in an Electric Field,  It takes energy (work) to move it.  This work can be done by an external force (you). FIELD negative  You can also think of this as the FIELD doing the negative of this amount of work on the particle.

And also remember: The net work done by a conservative (field) force on a particle moving around a closed path is ZERO!

IMPORTANT The work necessary for an external agent to move a charge from an initial point to a final point is INDEPENDENT OF THE PATH CHOSEN!

The Electric Field Is a conservative field.  No frictional losses, etc. Is created by charges. When one (external agent) moves a test charge from one point in a field to another, the external agent must do work. This work is equal to the increase in potential energy of the charge. It is also the NEGATIVE of the work done BY THE FIELD in moving the charge from the same points.

Electric Potential Energy When an electrostatic force acts between two or more charged particles, we can assign an ELECTRIC POTENTIAL ENERGY U to the system. The change in potential energy of a charge is the amount of work that is done by an external force in moving the charge from its initial position to its new position. It is the negative of the work done by the FIELD in moving the particle from the initial to the final position.

Definition – Potential Energy PE or U is the work done by an external agent in moving a charge from a REFERENCE POSITION to a different position. A Reference ZERO is placed at the most convenient position  Like the ground level in many gravitational potential energy problems.

Example: E Zero Level q F d Work by External Agent W external = F  d = qEd= U Work done by the Field is: W field = -qEd = -W external

AN IMPORTANT DEFINITION Just as the ELECTRIC FIELD was defined as the FORCE per UNIT CHARGE: We define ELECTRICAL POTENTIAL as the POTENTIAL ENERGY PER UNIT CHARGE: VECTOR SCALAR

UNITS OF POTENTIAL

Let’s move a charge from one point to another via an external force. The external force does work on the particle. The ELECTRIC FIELD also does work on the particle. We move the particle from point i to point f. The change in kinetic energy is equal to the work done by the applied forces. Assume this is zero for now.

Furthermore… If we move a particle through a potential difference of  V, the work from an external “person” necessary to do this is q  V

Consider Two Plates OOPS …

Look at the path issue

Important We defined an absolute level of potential. To do this, we needed to define a REFERENCE or ZERO level for potential. For a uniform field, it didn’t matter where we placed the reference. For POINT CHARGES, we will see shortly that we must place the level at infinity or the math gets very messy!

An Equipotential Surface is defined as a surface on which the potential is constant. It takes NO work to move a charged particle between two points at the same potential. The locus of all possible points that require NO WORK to move the charge to is actually a surface.

( )

S A B  V q ( ) S V

Field Lines are Perpendicular to the Equipotential Lines

Equipotential

Keep in Mind Force and Displacement are VECTORS! Potential is a SCALAR.

UNITS 1 VOLT = 1 Joule/Coulomb For the electric field, the units of N/C can be converted to: 1 (N/C) = 1 (N/C) x 1(V/(J/C)) x 1J/(1 NM) Or 1 N/C = 1 V/m So an acceptable unit for the electric field is now Volts/meter. N/C is still correct as well.

In Atomic Physics It is sometimes useful to define an energy in eV or electron volts. One eV is the additional energy that an proton charge would get if it were accelerated through a potential difference of one volt. 1 eV = e x 1V = (1.6 x C) x 1(J/C) = 1.6 x Joules.

Coulomb Stuff: A NEW REFERENCE: INFINITY Consider a unit charge (+) being brought from infinity to a distance r from a Charge q: q r To move a unit test charge from infinity to the point at a distance r from the charge q, the external force must do an amount of work that we now can calculate. x

Just Do It!

OK, doing it! Set the REFERENCE LEVEL OF POTENTIAL at INFINITY so (1/r A )=0.

For point charges

r1r1 r2r2 r3r3 P q1q1 q3q3 q2q2 (24 - 5)

For a DISTRIBUTION of charge:

dq O A (24 - 8)

A B V V+dV ( ) Calculating the electric field E from the potential V

A B V V+dV ( )

x y O q1q1 q2q2 q3q3 r 12 r 23 r 13 ( )

( ) O x y  q1q1 Step 1 x y O r 12 q1q1 q2q2  Step 2 Step 3 x y O r 12 r 13 r 23 q1q1 q2q2 q3q3 

path A B conductor ( )

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In the figure, point P is at the center of the rectangle. With V = 0 at infinity, what is the net electric potential in terms of q/d at P due to the six charged particles?

Continuing s