23. electric potential

Electrical energy

• Electrical energy is  an indispensable ingredient of our technological society.

• The concepts of potential and voltage are crucial to understanding how electric circuits work.

• Important applications include cancer radiotherapy, high-energy particle  accelerators, and others.

Mechanics

• The concepts of work, potential energy, and conservation of energy are extremely useful in our study of mechanics.

• These concepts are just as useful for understanding and analyzing electrical interactions.

Work Done

• When a force F acts on a particle that moves from point a to point b,

• The work done by the force is given by a line integral:

            W_a->b    =   _a∫^b  F . dl      =    _a∫^b  F cosф dl

• Work done by a conservative force is also the change in potential energy U.

            W_a->b    =  U_a  -  U_b    =   - ΔU

Work-Energy Theorem

• The work-energy theorem says that the change in kinetic energy during a displacement equals the total work done on the particle.

            K_a  +  U_a   =  K_b  +  U_b  

• The total mechanical energy, kinetic plus potential, is conserved only if conservative forces do work.

Electric Potential Energy in a Uniform Field

• The work done by the electric field is the product of the force magnitude and the component of displacement in the (downward) direction of the force.

            W_a->b   =  F d  =  q₀ E d

• The work is positive, since the force is in the same direction as the net displacement of the test charge.

Positive Charge moving in the Electric Field

• For a positive charge moving in the direction of the electric field, the potential energy decreases.

• For a positive charge moving in the direction opposite of the electric field, the potential energy increases.

Electric Potential Energy of Two Point Charges

• Coulomb’s law gives the force on charge

                  F  =  [ 1 / (4 π ε₀) ] [ q q₀ / r ² ]

It is proportional of  ( 1 / r ² ).

• Electric potential energy of two point charges

            U  =  [ 1 / (4 π ε₀) ] [ q q₀ / r ]

It is proportional of  ( 1 / r ).

Graphs of the Potential Energy

• If two point charges have the same sign, the interaction is repulsive, the work is positive, and the potential energy is positive.

• If two point charges have opposite sign, the interaction is attractive, the work is negative, and the potential energy is negative.

Electric Potential Energy with Several Point Charges

• Electrical potential energy of a point charge q₀ and collection of charges q₁, q₂, q₃, ...  is the algebraic sum of the contributions from the individual charges.

            U   =   [ q₀ / (4 π ε₀) ]   [  (q₁ / r₁) + (q₂ / r₂) + …. ]       =      [ q₀ / (4 π ε₀) ]    Σ_i [ q _i / r _i ]

Total Potential Energy

• For every electric field due to a static charge distribution, the force exerted by that field is conservative.

• The total potential energy is the sum of the potential energies of interaction for each pair of charges.

                  U  =   [ 1 / (4 π ε₀) ]    Σ_{i< j}  [ q_i q _j / r _ij ]

Electric Potential

• Potential is potential energy per unit charge.

            V  =  U / q₀      or          U  =  q₀ V

It is a scalar quantity. Similar to mileage, miles per gallon of gas!

• The SI unit of potential

                        1 V  =  1 volt  =  1 J/C

In honor of the Italian Alessandro Volta (1745-1827).

Voltage

• The voltage of a battery equals the difference in potential between its positive terminal (point a) and its negative terminal (point b).

             V_ab   =  U_a  -  U_b

• Voltage is the potential of a with respect to b, equals the work done by the electric force when a unit charge moves from a to b.

• A voltmeter is an instrument that measures the difference in potential between two points.

Calculating Electric Potential

• Electric potential due to a continuous distribution of charge is

                  V  =  [ 1 / (4 π ε₀) ]   ∫ dq  / r

• Electric potential difference

            V_a - V_b    =   W_a->b  / q₀

                                     =  ( 1 / q₀ )  _a∫^b  F . dl     =    ( 1 / q₀ ) _a∫^b  q₀ E . dl      =     _a∫^b  E cosф dl

• Moving with the direction of electric field means moving in the direction of decreasing potential.

Electric Potential and Electric Potential Energy

• Electric potential energy is measured in joules.

• Electric potential is potential energy per unit charge and is measured in volts.

• Note: The electric field always points from regions of high potential towards regions of low potential.  The direction of force on a point charge is in the direction of the electric field if the charge is positive but opposite of electric field is the charge is negative.

Electron Volts

• If a charge equals the magnitude of the electron charge and the potential difference is one volt, then the change in energy is defined to be 1 electron volt.

            1 eV  =  1.602 x 10^-19  J

The multiples meV, keV, MeV, GeV, TeV are often used.

• Electron volt is a unit of energy, not a unit of potential or potential difference.

• Problem (E23.3):  A proton (charge +e = 1.602 x 10^-19 C) moves a distance d = 0.50 m in a straight line between points a and b in a linear accelerator.  The electric field is uniform along this line, with magnitude E = 1.5 x 10⁷ N/C in the direction from a to b.  Determine the potential difference.   ( 7.5 MV )

Equipotential Surfaces

• An equipotential surface is a three-dimensional surface on which the electric potential is the same at every point.

• Equipotential surfaces for different potentials can never touch or intersect.

• Field lines and equipotential surfaces are always mutually perpendicular.

• When charges are at rest, a conducting surface is always an equipotential surface.

• Fields lines are perpendicular to a conducting surface.

• If the cavity contains no charge, every point in the cavity is at the same potential.

• The electric field is zero everywhere in the cavity, and there is no charge anywhere on the surface of the cavity.

• Gaussian surfaces have relevance only when we are using Gauss’s law, and we can choose any Gaussian surface that’s convenient.

• Equipotential surfaces cannot be chosen, and the shape is determined by the charge distribution.

Potential Gradient

• Electric field and potential are closely related.

                  V_a - V_b   =   _a∫^b  E . dl

The components of E are related to the partial derivatives of V with respect to x, y, and z.

• The electric field vector found from potential

            E  =   - [ i ( ∂V / ∂x )  +  j ( ∂V / ∂y )  +  k ( ∂V / ∂z ) ]

• In vector notion,

            E  =  - ∇V

E is the negative gradient of V or E equals negative grad V.

The quantity ∇V is called the potential gradient.