Electric Potential and Potential Energy -------------------------------------------------------- Joseph F. Alward, PhD     Department of  Physics   University of the Pacific

  

Positive   charges accelerate toward regions of lower potential.  Negative charges accelerate toward regions of higher potential.

  Equipotential Surfaces                                                 

Alejandro Volta  (1745-1827) ----------------------------------------- Points in an electric field may be labeled according to their "potential".

Imaginary concentric spheres surrounding a positive point charge are surfaces of equal potential:  8 volts, 5 volts, etc. Equipotential surfaces are perpendicular to the electric field lines.

 
 Potentials and Charge
 Distributions   

Potentials are more positive in regions which are more positive, or less negative.

 Equipotential Surfaces                                

Equipotential surfaces are 3-dimensional.

Equipotential Surfaces in a Constant Electric Field      

Equipotential surfaces are planes  perpendicular to the electric field lines.  (2-dimensional view)

3-dimensional view Equipotential surfaces are imaginary planes.

  An Equipotential Surface Near Two Opposite Charges

  Equipotential Surfaces and Electric Fields                        

Cross-sectional view of 3-dimensional equipotential surfaces.  Equipotential surfaces are perpendicular to the electric field lines.

Equipotential surfaces (broken lines) are perpendicular to the electric field.

                                                                                             

Positive   charges accelerate toward regions of lower potential.  Negative charges accelerate toward regions of higher potential.

  Potential Differences and
  Equipotential Surfaces  

VA = 500 V       VB =  100 V   VC =  100 V

DVBA = VB - VA            = 100 V - 500 V            = - 400 V ------------------------------- DVAB = VA - VB            = 500 V - 100 V            = 400 V ------------------------------- DVCB = VC - VB            = 100 V - 100 V            = 0

   Definition of Potential Difference              

W = work done by E DV = -W/q     DV =  -8 J /2 C       = - 4 V

Calculating E from Potential Difference    

W  = -q DV         W =     F Ds              = (qE)Ds   (qE)Ds = -q DV    E =  -DV / Ds ------------------------ What is the sign of DV ? If E is positive, E points in the direction of  Ds.

   Calculating E from Potential Difference               

What is E between the plates?

DV = VB - VA        = 12 - 0        = 12 V Ds   = 0.02 m E     = - DV / Ds        = -12 /0.02        = -600 V/m -------------------- E is negative, so it points opposite to Ds:  E points from B to A.

    Calculating E from Potentials

What is E near top of this person's    head, at the 100 V equipotential    surface?

Person in contact with ground is at same potential as ground:  V = 0. Point is roughly midway between 200 V and 0 V surfaces. Estimated separation:   Ds = 30 cm                                         = 0.30 m E    = -DV / Ds       =  -(0 - 200)/0.30       = 667 V/m

  Calculating E from Potentials

What is the electric field E?

E    = -DV / Ds       DV = 2 V Ds = 1 x 10-3 m E = 2000 V/m ---------------- Note:  E always points toward lower potentials.

                                                                                               

Positive   charges accelerate toward regions of lower potential.  Negative charges accelerate toward regions of higher potential.

 E Points Toward Lower Potentials
  Positive    Charges Move Toward Lower Potentials
  Negative Charges Move Toward Higher Potentials                    

Electric field arrows point toward regions of lower potential.

  Metals Are Equipotential Surfaces                                           

How much work would it take to drag over the surface a positive charge q from Point A to Point B? --------------------------------------------------------------- Moving along the surface, DV = 0, so E//    = -DV / Ds         = 0 E// = component of E parallel to the surface so the electric force F// = qE// = 0. W = F// Ds      = 0

                                                                                                  

No work is required to move charges along equipotential surfaces.

  Heart Equipotential Surfaces

Which way is the negative ion current flowing across this heart? The positive ion current?

   The Potential Due to a Single Point Charge                               

What is the potential 2 meters away  from a one nano-coulomb charge?

V = V(r) = kq/r   This expression for potential is only   valid for a single point charge.  ------------------------------------------------- k = 9 x 109 (SI units) q = 1 x 10-9 C  r = 2 m V = 9 x 109 (1 x 10-9) / 2     = 4.5 volts     = 4.5 V ------------------------------------------------- What is the potential at any point a distance r = 1.5 meter from the charge?             Answer:  6 V

    Work and Potential                                                                        

How much work would be done by   the electric field if Q = 20 C were   moved from A to B?

W   = -QDV  Q   = 20 C DV  = 8V - 4.5 V         = 3.5 V W = -(20 C) (3.5 V)      = -70 J    (note:  negative) ---------------------------------------- How much work was done by the agent which moved the charge? Answer:  +70 J

   

Positive   charges accelerate toward regions of lower potential.  Negative charges accelerate toward regions of higher potential.

  Breakdown of Air                                   

  Lightning Energy                                                                       

A charge q = - 30 C is pushed  from cloud to ground.   DV =  2 x 108 V.

Energy gained = Work done by electric field                       = -qDV                       = -(-30 C)(2 x 108 V)                       = 60 x 108 J ------------------------------------------------------------ How many kilograms of water at 20 C could be converted into steam at 100 C? (For help with this, click here to go to the lecture on specific and latent heats.) Answer:  2308 kg (2.3 cubic meters) (What would this lightning bolt do to a human?)

  Calculate the Potential Due to Two Point Charges     

What is the potential at Point P?

Use V = kq/r for each charge, then add the potentials: V = (9 x 109)(4 x 10-9)/12     = 4 V V = (9 x 109)(6 x 10-9)/27     = 2 V Total Potential:  6 V ----------------------------------------  How much work was done to  assemble these charges?

   Work Done in Assembling a System of Charges         

How much work would you have to do to assemble this system of charges?

The work required to bring the first charge (4 nC) in from infinity is zero. The work done by the electric field when the 6 nC charge is brought in is W =  - (6 nC) DV.  The work done by some external agent (you) is the negative of this work: Wyou  = (6 nC) DV. DV = ?

  The Electric Potential Energy of a Charge Configuration    

V =  (9 x 109)(4 x 10-9)/36     = 1 V  DV = 1 V - 0 V        = 1 V   W = qDV        = (6 nC) (1 V)        = 6 nJ  This is the work done by an external  agent to assemble this system of  charges; it's not the work done by  the electric field.     This is the "electric potential energy" of this system.

 Potential Energy of Three
 Charges on a Line 

Solution:  Bring in first charge for free.  Second charge sees the potential field of the first.  Third charge sees the sum of the two potential fields.

                                                                                                    

No work is required to move charges along equipotential surfaces.

   Conservation of Energy Example                              

Energy = kinetic + potential      E    = 1/2 mv2 + qV            = KE + EPE ------------------------------------  EB = EA  1/2 mvB2 + qVB = 1/2 mvA2 + qVA  vA= 0  vB = [2q(VA - VB)/m]1/2  Solve for speed v:      v = 5 m/s

  Electrocardiography                               

  Electroencephalography                        

Greek:   enkephalos:  in the head; brain.  The alpha rhythm is the "resting" rhythm.

  Electroretinography                                    

Measuring the electrical activity of the retina.

    The Potential Field of a Blood Drop                

Normal                       Osteosarcoma              Leukemia  Electrosensitive liquid crystal changes color with voltage.  Blue:  150 V     Green:  120 V     Yellow:  90 V     Red:  50 V

  Potential Field of Brain     

A map of the equipotentials in the brain of a person with epilepsy.

 
   300,000 Volts and Still Alive      

Electrons flow briefly onto this person's body, lowering her potential to -300,000 volts, the same as the wire's potential. How close could her hand or foot come to the tower (which is grounded, V = 0) without her being electrocuted?

  Conceptual Questions:

Base your answers to the following questions on the equation:  E    = -DV / Ds. ---------------------------------------------------------------------------------------------------------------- 1.  If the potential in a region of space is constant, is the electric field necessarily      zero in the region? 2.  If the electric field is zero in a region of space, must DV be zero between points       in that region? 3.  If E is zero in a region of space, must the potential V be zero at all points in that      region, too?