The Potential Difference

Charged particles exert forces on each other.  The electric field E = F/q produced by a charged particle at some position r in space is a measure of the force F the particle exerts on a test charge q, if we place the test charge at r.  The electric field E is a vector.  The electric field due to a charge distribution is the vector sum of the fields produced by the charges making up the distribution.  When we think about electricity in everyday life, we seldom think about the electric fields.  We are more familiar with the concepts of voltage, current, and power. 

How is the voltage related to the electric field?

When a particle with charge q is placed in an external electric field E, (i.e.  an electric field produced by other charges), then an electric force F = qE will act on it.  If this force is not balanced by other forces, then the particle will accelerate, and its kinetic energy will change.  The electric force will do positive work on the particle.  If the electric force is balanced by another external force Fext = -qE, and this external force moves the particle against the electric force, than the external force Fext does positive work.  The work done is

W = ∫Fextdr = -q∫Edr.

The work done by the external force is equal to the change in the electrostatic potential energy of the particle in the external field.  The change in the potential energy of a charge q when being moved from point A to point B, is the work done by an external force in moving the charge.

∆U = UB - UA = -q ∫AB Edr.

The integral is taken along a particular path.  But the electrostatic force is a conservative force.  The integral is independent of the path.  ΔU therefore depends only on the endpoints A and B of the path, not on the actual path itself.

The change in potential energy is proportional to the charge q.  Its sign depends on whether the charge is positive or negative.  We define the potential difference or voltage ΔV as the potential energy difference divided by the charge, or the potential energy difference per unit charge.  The potential difference is the potential energy difference of a small, positive test charge, divided by the charge.  As in the case of gravity, the zero of the potential energy and therefore the zero of the potential are not uniquely defined.  Only potential energy differences and potential differences are unique.  The SI unit of energy is J, therefore the SI unit of potential is J/C.  We define Volt as V = J/C.  The electrostatic potential is a scalar, not a vector. 

We say that a charge distribution, which produces an electric field, also produces an electric potential.  The electrostatic potential produced by a finite charge distribution is, by convention, set to zero at infinity.  Then the potential V(r) of the distribution is the work done per unit charge in bringing a small test charge from infinity to position r.  For a charge distributions which extend to infinity, we cannot set the zero of the potential at infinity, because then the potential would be infinite everywhere, and it would be a useless concept.  We then set the zero at some convenient reference point, but we always must specify the reference point along with the potential, since there is no unique convention.  For many applications we choose the ground to be the zero of the potential.

The potential difference between the two terminals of an A, B, C, or D cell battery is 1.5V.  For every Coulomb of negative charge that is moved from the positive to the negative terminal, 1.5 J of work must be done against electric forces, and 1.5 J of some other form of energy is converted into electrostatic energy.  When the terminals are connected by a wire, then the charge is free to move inside the wire, and the electric field does work on the charge.  This work can now be reconverted into some other form of energy.

 

The potential at a point r of a positive point charge located at the origin is the work that must be done per unit charge in bringing a test charge from infinity to r.

V(r) = -∫r Edr = ∫rEdr.

We can bring the test charge along an arbitrary path, which we can think of as being made up of infinitesimal small steps, either in the radial direction or perpendicular to the radial direction.  We do not have to do work when we step perpendicular to the radial direction, because then E is perpendicular to dr.  Along the radial direction Edr = Edr, because E and r points both point outward.  The integral therefore becomes

V(r) = -∫r Edr = ∫rEdr = ∫r(keq/r2)dr = keq/r = q/(4πε0r).

The potential due to a point charge is proportional to the inverse of the distance from the point charge.

To find the potential due to a collection of charges, we use the principle of superposition and add the potentials due to the individual charges.  Because the potential is a scalar, and not a vector, we just have to add numbers.

Just as we described the electric field around a charged object by field lines, we can also describe the electric potential pictorially with equipotential surfaces (contour plots).  Each surface corresponds to a different fixed value of the potential.  Equipotential lines are lines connecting points of the same potential.  They often appear on field line diagrams.  Equipotential lines are always perpendicular to field lines, and therefore perpendicular to the force experienced by a charge in the field.  If a charge moves along an equipotential line, no work is done; if a charge moves between equipotential lines, work is done. 

Example:

Field lines and equipotential lines for a positive point charge are shown below.

 

Problem:

In the Bohr model of the hydrogen atom, the electron orbits the proton at a distance of r = 5.29*10-11 m.  How much work must be done to completely separate the electron and the proton?

bulletSolution:
The force on the electron is the Coulomb force between the proton and the electron.  It pulls the electron towards the proton.  For the electron to move in a circular orbit, the Coulomb force must equal the centripetal force. 
We need keqe2/r2 = mv2/r.
This yields mv2 = keqe2/r, so the kinetic energy of the electron is
KE(r) = mv2 = keqe2/r. 
The potential energy of the electron in the field of the positive proton point charge is U(r) = -eV(r) = - keqe2/r.  The total energy is the sum of the electron's kinetic energy and its potential energy.
KE(r) + PE(r) = -keqe2/r = (-1/2 (9*109)(1.60*10-19) /(5.29*10-11) J = -2.18*10-18 J. 
This is usually stated in energy units of electron volts (eV).  An eV is 1.6*10-19 J, so dividing by this gives an energy of -13.6 eV.  To remove the electron from the atom, i.e. to move it very far away and give it zero kinetic energy, 13.6 eV of work must be done by an external force.  13.6 eV is the ionization energy hydrogen. 

Problem:

An alpha particle containing two protons is shot directly towards a platinum nucleus containing 78 protons from a very large distance with a kinetic energy of 1.7*10-12 J.  What will be the distance of closest approach?

bulletSolution:
The electric charge of the alpha particle is q1 = 2 qe and that of the platinum nucleus is q2 = 78 qe.  The alpha particle and the nucleus repel each other.  As the alpha particle moves towards the nucleus, some of its kinetic energy will be converted into electrostatic potential energy.  At the distance of closest approach, the alpha particle's velocity is zero, and all its initial kinetic energy has been converted into electrostatic potential energy. 
The initial kinetic energy of the alpha particle is KE = 1.7*10-12 J. 
The final potential energy is U = kq1q2/d, where d is the distance of closest approach. 
We have keq1q2/d = 1.7*10-12 J.  This yields
d = keq1q2/(1.7*10-12 J) = (9*109*2*78*(1.6*10-19)2/(1.7*10-12)) m = 2.1*10-14 m
for the distance of closest approach.  The alpha particle's velocity is zero at d.  The acceleration is in a direction away from the nucleus.  The distance between the alpha particle and the nucleus will increase again, converting the potential energy back into kinetic energy. 

The electrostatic potential V is related to the electrostatic field E.  If the electric field E is known, the electrostatic potential V can be obtained using V = -∫Edr.
How can we obtain the field from the potential?

 

Consider the two points P1 and P2 shown in figure above.  Assume that they are separated by an infinitesimal distance dL.  The change in the electrostatic potential between P1 and P2 is given by dV = -EdL = -EdLcosθHere θ is the angle between the direction of the electric field and the direction of the displacement dL.  We can rewrite this equation as

dV/dL = -Ecosθ = -EL.

EL indicates the component of the electric field along the direction of dL.  If the direction of the displacement is chosen to coincide with the x-axis, this becomes

dV/dx = -Ex.

For the displacements along the y-axis and z-axis we obtain dV/dy = -Ey and dV/dz = -Ez.

The total electric field E can be obtained from the electrostatic potential V by combining these three equations.

We say that E is the gradient of the potential V.

Note: The derivatives above are partial derivatives.  When taking the derivative of V with respect to x, we assume that y and z are constant, when taking the derivative of V with respect to y we assume that x and z are constant, and when taking the derivative of V with respect to z we assume that x and y are constant.

In many electrostatic problems the electric field due to a certain charge distribution must be evaluated.  The calculation of the electric field can be carried out using two different methods:

bulletThe electric field can be calculated by applying Coulomb's law and vector addition of the contributions from all charges of the charge distribution.
bulletThe total electrostatic potential V can be obtained from the algebraic sum of the potential due to all charges that make up the charge distribution, and the field can be found by calculating the gradient of V.

In many cases the second method is simpler, because the calculation of the electrostatic potential involves an algebraic sum, while the direct calculation of the field involves a vector sum. 

Problem:

In some region of space, the electrostatic potential is V(x,y,z) = x2 + 2xy.  Find E(x,y,z). 

bulletSolution:

Ex = -∂V/∂x = -2x - 2y.  Ey = -∂V/∂y = -2x.  Ez = -∂V/∂z = 0.

Exercise  (You can earn up to 5 points extra credit by completing this exercise.)

A system consisting of positive and a negative charge of equal magnitude q, separated by a distance d is called an electric dipole.  For an electric dipole we define a new vector, called the electric dipole moment.  The magnitude of the dipole moment vector p is the magnitude of the charge q times the distance d between them, p = qd.  The vector points from the negative towards the positive charge.

The dipole moment is a useful concept when effects of microscopic charge separation are observable, but the actual distance between the charges is too small to be measured.  Molecules can have permanent dipole moments and atoms and molecules without a permanent dipole moment can acquire one when placed in an external electric field. 

The potential produced by an electric dipole is calculated summing the potential of the two point charges that produce it.  For a point r whose distance from the negative charge is r- and from the positive charge is r+ we have

V(r) = kq[1/r+ - 1/r-] = kq(r- - r+)/(r+r-).

If a the negative charge is placed on the z-axis at z = -d/2 and the positive charge is placed on the z-axis at z = +d/2 then for large r this becomes

V(r) = kqd cosθ/r2 = kpcosθ/r2,

where r- - r+ = ∆r = d cosθ, and 1/(r+r-) is nearly equal to 1/r2, since r >> ∆r. 

The dipole potential does not have spherical symmetry and decreases as 1/distance2, much faster than the Coulomb potential, which decreases as 1/distance.  The dipole field decreases as 1/distance3, and dipole effects become quickly negligible as the distance increases.  The figure to the right shows the equipotential lines and field lines of an electric dipole. 

In a uniform electric field the net force on an electric dipole is zero.

But the dipole moment p is not aligned with the electric field, then a torque acts on the dipole, trying to align p with E.  The magnitude of this torque if τ = pE sinθ.

To rotate the dipole away from alignment, you have to apply an external torque and do work.  The work done by this external torque is stored as potential energy and can be converted into other forms of energy.  The potential energy of a dipole in an external field is

Up = -pE cosθ.

The potential energy is lowest when the dipole is aligned with E and highest if it is anti-aligned. 

If the field is not uniform, then the magnitude of the electric force acting on the positive charge can be different from that acting on the negative charge, and there can be a net force acting on the dipole.

Please complete assignment 10 now.