Lecture 11
Potential Difference and Electric Potential
When the test charge is moved in the field by some external agent, the work done by the field on the charge is equal to the negative of the work done by the external agent causing the displacement. This is analogous to the situation of lifting an object with mass in a gravitational field—the work done by the external agent is mgh and the work done by the gravitational force is - mgh.
For an infinitesimal displacement ds of a charge, the work done by the electric
field on the charge is F. ds = q0E. ds. As this amount of work is done by the field, the potential energy of the charge –field system is changed by an amount dU =-q0E.ds. For a finite displacement of the charge from point A to point B, the change in potential energy of the system ΔU = UB - UA is
(1)
The integration is performed along the path that q0 follows as it moves from A to B. Because the force q0E is conservative, this line integral does not depend on the path taken from A to B.
The potential energy per unit charge U/q0 is independent of the value of q0 and has a value at every point in an electric field. This quantity U/q0 is called the electric potential (or simply the potential) V. Thus, the electric potential at any point in an electric field is
(2)
The fact that potential energy is a scalar quantity means that electric potential also is a scalar quantity.
As described by Equation 1, if the test charge is moved between two positions A and B in an electric field, the charge–field system experiences a change in potential
energy. The potential difference ΔV = VB- VA between two points A and B in an
electric field is defined as the change in potential energy of the system when a test
charge is moved between the points divided by the test charge q0:
(3)
Potential difference should not be confused with difference in potential energy.
The potential difference between A and B depends only on the source charge distribution (consider points A and B without the presence of the test charge), while the difference in potential energy exists only if a test charge is moved between the points. Electric potential is a scalar characteristic of an electric field, independent of any charges that may be placed in the field.
From Equation 3, the work done by an external agent in moving a charge q through an electric field at constant velocity is
(4)
Because electric potential is a measure of potential energy per unit charge, the SI
unit of both electric potential and potential difference is joules per coulomb, which is defined as a volt (V):
That is, 1 J of work must be done to move a 1- C charge through a potential difference of 1 V.
Equation 3 shows that potential difference also has units of electric field times
distance. From this, it follows that the SI unit of electric field (N/C) can also be
expressed in volts per meter:
Therefore, we can interpret the electric field as a measure of the rate of change
with position of the electric potential.
A unit of energy commonly used in atomic and nuclear physics is the electron volt
(eV), which is defined as the energy a charge–field system gains or loses when
a charge of magnitude e (that is, an electron or a proton) is moved through a
potential difference of 1 V. Because 1 V =1 J/C and because the fundamental
charge is 1.60 x 10-19 C, the electron volt is related to the joule as follows:
(5)
The Electron Volt
The electron volt is a unit of energy, NOT of potential. The energy of any system may be expressed in eV, but this unit is most convenient for describing the emission and absorption of visible light from atoms. Energies of nuclear processes are often expressed in MeV.
Potential Differences in a Uniform Electric Field
Because E is constant, we can remove it from the integral sign; this gives
(6)
The negative sign indicates that the electric potential at point B is lower than at point A; that is, VB VA. Electric field lines always point in the direction of decreasing electric potential, as shown in Figure 25.2a.
Figure 1 (a) When the electric field E is directed downward, point B is at a lower
electric potential than point A. When a positive test charge moves from point A to
point B, the charge–field system loses electric potential energy. (b) When an object of
mass m moves downward in the direction of the gravitational field g, the object–field
system loses gravitational potential energy.
Obtaining the Value of the Electric Field from the Electric Potential
The electric field E and the electric potential V are related as shown in Equation 3.
We now show how to calculate the value of the electric field if the electric potential is known in a certain region.
From Equation.3 we can express the potential difference dV between two points a distance ds apart as
(7)
If the electric field has only one component Ex, then E. ds = Ex dx. Therefore, Equation 25.15 becomes dV = -Ex dx, or
(8)
When a test charge undergoes a displacement ds along an equipotential surface, then dV = 0 because the potential is constant along an equipotential surface. From Equation 7, we see that dV=-E.ds = 0; thus, E must be perpendicular to the displacement along the equipotential surface. This shows that the equipotential surfaces must always be perpendicular to the electric field lines passing through them.
Figure 2. Equipotential surfaces (the dashed blue lines are intersections of these
surfaces with the page) and electric field lines (red-brown lines) for (a) a uniform
electric field produced by an infinite sheet of charge, (b) a point charge, and (c) an
electric dipole. In all cases, the equipotential surfaces are perpendicular to the electric
field lines at every point.
Definition of Capacitance
Consider two conductors carrying charges of equal magnitude and opposite sign, as shown in Figure 26.1. Such a combination of two conductors is called a capacitor. The conductors are called plates. A potential difference !V exists between the conductors due to the presence of the charges.
Experiments show that the quantity of charge Q on a capacitor1 is linearly proportional to the potential difference between the conductors;
The capacitance C of a capacitor is defined as the ratio of the magnitude of the
charge on either conductor to the magnitude of the potential difference between
the conductors:
(9)
Figure 3 A capacitor consists of
two conductors. When the capacitor
is charged, the conductors carry
charges of equal magnitude and
opposite sign.
Note that by definition capacitance is always a positive quantity. Furthermore, the charge Q and the potential difference ΔV are always expressed in Equation 9 as positive quantities.
Capacitance has SI units of coulombs per volt. The SI unit of capacitance is the farad (F), which was named in honor of Michael Faraday:
The farad is a very large unit of capacitance. In practice, typical devices have capacitances ranging from microfarads (10-6 F) to picofarads (10-12 F). We shall use the symbol µF to represent microfarads.
The capacitance of an isolated charged sphere is proportional to its radius and is independent of both the charge on the sphere and the potential difference
(10)
The capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation
(11)
The equivalent capacitance of a parallel combination of capacitors is the algebraic sum of the individual capacitances and is greater than any of the individual capacitances.
(12)
The inverse of the equivalent capacitance is the algebraic sum of the
inverses of the individual capacitances and the equivalent capacitance of a series
combination is always less than any individual capacitance in the combination.
(13)
Energy Stored in a Charged Capacitor
The potential energy stored in a charged capacitor in the following forms:
(14)