Chapter 29 – The Electric Potential

 

This chapter introduces the concept of Electric Potential which is closely related to Electric Field.  The electric field was defined as a quantity that is produced at a point in space by source charges.  The definition was E = F/q where q was a “little” charge and F was the force at a particular place.  The electric field at that place was just F/q.

 

The Electric Potential is defined analogously as V = Uelectric/q, where Uelectric is the electric potential energy of the charge q at a particular place.  The electric potential energy depends on the location of all the other charges interacting with q.  The electric potential, V, at that place is just Uelectric/q.  Once we know V, we don’t have to know anything about the source charges that produced V.  Notice the strong similarity between electric field and electric potential.  They are both defined per Coulomb, electric field is Newtons per Coulomb and electric potential is Joules per Coulomb.

 

The connection is even stronger because the electric field can be found by differentiating the electric field, Ex = - dV/dx, Ey = - dV/dy, and Ez = - dV/dz.  That is the vector components of the electric field can be found by taking the appropriate derivatives of the electric potential.  This operation is called the gradient, E = - gradient of V.  This gives us a new way to find electric fields.  In general, it is easier to calculate a scalar quantity, V, than a vector quantity E.  Consequently, it is often more convenient to find V first and then differentiate it to find the different components of the vector electric field.

 

One last point to keep in mind before we get bogged down in details is that the units of electric potential are Volts, the same volts that are listed on batteries and are piped into our homes to power our appliances.  Consequently it will be real important to understand the difference between electric potential, V, and electric potential energy, U.

 

1.         Let’s begin by reviewing the conservation of mechanical energy.  For a system with “conservative” forces acting between its various parts,

 

            Emechanical = Kinetic Energy + Potential Energy.

 

This is useful because in many situations the total mechanical energy is conserved, Δ Emechanical = 0,

 

            KEfinal + Ufinal = KEinitial + Uinitial, where U is the potential energy.

 

2.         Imagine a bowling ball sitting on a table.  Gravity exerts a downward force which is balanced by the upward force exerted by the table, the thing we call the normal force.  It is called the normal force because it is perpendicular to the table and not because it is less strange than other forces!  Back to the bowling ball.  Nothing is happening to the ball.  The net force is zero and the ball is quietly sitting on the table.  Now I decide to lift the ball very carefully be applying an upward force exactly equal to the force of gravity.  In fact I have to apply a force infinitesimally larger than the gravitational force to get it moving.  But my goal is to move the ball a distance y above the table while keeping the net force equal to zero.  No work is done on the bowling ball.

 

            Net Work = ∫Fnet∙ ds = 0 since Fnet = 0.

 

From PHYS 180, the net work equals the change of kinetic energy and since I moved the bowling ball very carefully so that it’s velocity was zero, there was no change of kinetic energy of the bowling ball.

 

But my arms tell me that I have done serious work in lifting the bowling ball.  Where has the work that I have done gone?  It goes into increasing the potential energy of the ball-earth system, that is I have created gravitational potential energy!

 

Work done by me = ∫Fme∙ ds = Δ U = Ufinal - Uinitial. 

 

Notice that the force I exert in lifting the ball is in the opposite direction of the gravitational force and in the same direction as ds, so I do positive work and Ufinal is greater than Uinitial.  Lastly, notice that the gravitational force is just equal and opposite to the force I exert, so in terms of the work done by the gravitational force,

 

Work done by the force of gravity = ∫Fgravity∙ ds =  -Δ U = -Ufinal + Uinitial. 

 

This is the equation that is used to find the potential energy for a particular force.

 

3.         Let’s use that definition to find the change in potential energy in lifting the ball from yi to yf, where “i” and “f” stand for initial and final.  If yf > yi, then the force is in the opposite direction of ds and the work done by gravity is - mg(yf - yi) = -mgyf + mgyi = -Ufinal + Uinitial.  So we can use this to identify the gravitational potential energy as mgy near the earth’s surface.  Note that the only thing we care about is the difference between the potential energy at one spot and the potential energy at another spot.  So I can choose any convenient height y as the place where the gravitational potential energy is zero.  This is a general property of potential energy.  We can define the zero of potential to be any convenient location.  The location is generally selected to “simplify” calculations.

 

4.         As one last example from PHYS 180,  let’s find the potential energy in a mass – spring system.  The force exerted by a spring is equal to k x, where k is the spring constant and x is the distance the mass is pulled from its equilibrium position.  If we are stretching the spring from some xi to xf, the force exerted by the spring is opposed to the stretch.  The work done by the spring is just,

 

Work done by spring force = ∫Fspring∙ ds = - k ∫x dx integrated from xi to xf  which is equal to -ΔU = -Ufinal + Uinitial.  The integral is just x2/2.  The result shows that for a spring - mass system, Uspring = ½ kx2.

 

5.         Now let’s begin to apply this stuff to the material in PHYS 181.  Inside a capacitor, a positive charge q feels a force qE towards the negative plate of the capacitor.  In moving a small distance Δx toward the negative plate, the electric force does qE Δx amount of work,

 

            qE Δx = qE (xf - xi) =  -ΔU = -Ufinal + Uinitial

 

Starting at the positive plate, xi = 0, and ending on the negative plate, xf = d, where d is the separation between plates, and picking Ufinal, the electric potential energy, to be zero at the negative plate, we get,  qEd = Uinitial . 

 

Now use the definition of electric potential, V = U/q, to get to the main point of this little exercise, namely that the potential at the positive plate of a parallel plate capacitor is Ed and the potential at the other side is zero.  Also the potential drops linearly going from the positive to the negative plate.

 

This makes perfectly good sense in terms of energy.  If a positive charge is released with zero kinetic energy at the positive plate, it accelerates towards the negative plate losing electric potential energy and gaining kinetic energy.  At the negative plate, the conversion is complete,

 

qEd = ½ mv2.

 

6.         Next we find the electric potential due to a single point charge.  We do this by finding the electric potential energy of a pair of charges and then divide by the charge of the “source.”  Again notice that this is analogous with the way we found the electric field due to a point charge.

 

∫Fcoulomb∙ ds =  -Δ U = -Ufinal + Uinitial = k ∫(q1q2/x2)dx     

 

For concreteness, assume both charges are positive and the distance between the charges is increasing.  Then the electric force and ds are parallel and the above equation has the correct sign.  The integral is -1/x, so we get,

 

-kq1q2/xf + kq1q2/xi = -Ufinal + Uinitial .  This let’s us identify the electric potential energy of two charges as U = kq1q2/x.  This makes sense because two positive charges have the most potential energy when they are close together and the least when they are far apart.  U has exactly that behavior.

 

Though the equation was derived for two positive charges it works for any pair of charges.  Note that the equation “automatically” establishes a zero of potential energy at x equals infinity, when the charges are infinitely far apart.

 

The electric potential, V, established in space about the charge q2 is just U/q1, or V = kq2/x.  The analogy between electric field and electric potential is very strong since E = F/q and V = U/q but V is a scalar while E is a vector.  In general, this makes V much easier to calculate.

 

7.         The electric potential due to an array of charges, discrete or continuous, is found the same way that the electric field was found for an array of charges.  Namely, for a discrete set of charges, we add the potential due to each individual charge.  Note that since potential is not a vector, this sum is much easier.  For a continuous distribution of charges, break the distribution into little chunks of charge, sum up the contribution from all the little chunks, and then turn the sum into an integral.

 

8.         The energy graphs in the chapter, those that plot potential energy versus distance and then draw a horizontal line representing the constant total energy are very, very useful for getting a qualitative sense of what happens under various situations.  Total Energy = Kinetic Energy  + Potential Energy, E = KE + U, so KE = E – U which is just the separation between the total energy and potential energy lines on the energy graph.  Therefore the graph gives a very nice representation of the motion of a particle with a changing amount of electric potential energy.  The larger the separation, the faster the particle moves.  When the separation equals zero, kinetic energy is zero, and the velocity is zero.  Those points where E intersects U(x) are called turning points because they represent the limits of the allowed motion for the particle.  That is the particle cannot move into regions where E – U < 0 since a negative KE implies an imaginary velocity!