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.

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 **= q**E** 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 **F**_{ext}= -q**E**,
and this external force moves the particle against the electric force, than the external
force **F**_{ext} does positive work.

Recall!

- The work W done on an object by a constant force is defined as W =
**F**∙**d**= F d cosθ. Here F is the magnitude of the force, d is the magnitude of the displacement vector, and θ is the angle between the directions of the force and displacement vectors.

The work done by a varying force in one dimension is defined as

W = lim_{∆x-->0}Σ_{xi}^{xf}F_{x}∆x = ∫F_{x}dx.

In three dimensions we write

W = lim_{∆r-->0}Σ^{ }**F**∙∆**r**= ∫**F**·d**r**= ∫ F_{x}·dx + ∫F_{y}·dy + ∫F_{z}·dz.

Therefore the work done done by an external force balancing the electric force is

W = ∫**F**_{ext}·d**r**= -q∫**E**·d**r**.

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 = U_{B} - U_{A} = -q ∫_{A}^{B}**E**·d**r**.

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** ∆V as the
potential energy difference divided by the charge, or the potential energy
difference per unit (positive) 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 1.5 V. 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
}**E**·d**r** = ∫_{r}^{∞}**E**·d**r.**

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 d**r**. Along the radial direction
**E**·d**r **= Edr, because **E** and **r** points
both point outward. The integral therefore becomes

V(**r**) = ∫_{r}^{∞}Edr
= ∫_{r}^{∞}(k_{e}q/r^{2})dr
= k_{e}q/r = q/(4πε_{0}r).

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

If the point charge is not located at the origin, but at position **r**', we write

V(**r**) = kq/|**r** - **r**'|
= kq/((x - x'^{2}) + (y - y'^{2}) + (z - z'^{2}))^{1/2}

The potential outside a spherically symmetric charge distribution with total charge q is the same as that of a point charge q. 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.

Equipotential lines are 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.

Field lines and equipotential lines for a positive point charge are shown in the figure.

In the Bohr model of the hydrogen atom, the electron orbits the proton
at a distance of r = 5.29*10^{-11 }m. The proton has charge +q_{e}
and the electron has charge -q_{e}, where q_{e} = 1.6*10^{-19}
C. How much work must be done
to completely separate the electron and the proton

Solution:

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 k_{e}q_{e}^{2}/r^{2
}= mv^{2}/r.

This yields mv^{2
}= k_{e}q_{e}^{2}/r, so the
kinetic energy of the electron is

KE(r) = ½mv^{2 }= ½k_{e}q_{e}^{2}/r.

The potential energy of the electron in the field of the positive proton
point charge is U(r) = -q_{e}V(r) = - k_{e}q_{e}^{2}/r.

The
total energy is the sum of the electron's kinetic energy and its
potential energy.

KE(r) + PE(r) = -½k_{e}q_{e}^{2}/r = (-½) (9*10^{9})(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).
1 eV =
1.60*10^{-19 }J.

-2.18*10^{-18} J * 1eV/(1.60*10^{-19
}J) = -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.

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?

Solution:

The electric charge of the alpha particle is q_{1 }= 2q_{e} and
that of the platinum nucleus is q_{2 }= 78 q_{e}. 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 = kq_{1}q_{2}/d, where d
is the distance of closest approach.

We have k_{e}q_{1}q_{2}/d = 1.7*10^{-12
}J. This yields

d = k_{e}q_{1}q_{2}/(1.7*10^{-12
}J)
= (9*10^{9}*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.

An evacuated tube uses a voltage of 5 kV to accelerate electrons from rest to
hit a phosphor screen.

(a) How much kinetic energy does each electron gain?

(b) What is the speed of an electron when it hits the copper plate?

Solution:

(a) Each electron loses 5 keV of potential energy and
gains 5 keV = (5000 eV)(1.6*10^{-19} J/eV) =
8*10^{-16} J of kinetic energy.

(b) E = ½m_{e}v^{2}. v^{2} = 2E/m_{e}
= (2*8*10^{-16} J)/(9.1*10^{-31} kg) = 1.75*10^{15}
(m/s)^{2}.

v = 4.2*10^{7} m/s. This is more than 1/10 the speed of light.

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 = -∫**E**·d**r**.

Consider the two points P_{1} and P_{2} shown in figure
above. Assume that they are separated by an infinitesimal distance d**L**. The change in the electrostatic potential between P_{1} and P_{2}
is given by dV = -**E**∙d**L **= -E dL cosθ.**
**Here θ is the angle between the direction of the electric field and the
direction of the displacement vector d**L**. We can rewrite
this equation as

dV/dL = -E cosθ = -E_{L}.

E_{L} indicates the component of the electric field along the
direction of d**L**. If the direction of the displacement is chosen to
coincide with the x-axis, this becomes

dV/dx = -E_{x}.

For the displacements along the y-axis and z-axis we obtain dV/dy = -E_{y}
and dV/dz = -E_{z}.

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.

**E** = -(i∂E_{x}/∂x
+ j∂E_{y}/∂y + k∂E_{y}/∂y) = -**∇**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:

- The electric field can be calculated by applying Coulomb's law and vector addition of the contributions from all charges in the charge distribution.
- The 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.

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

Solution:

E_{x }= -∂V/∂x = -2x - 2y. E_{y
}= -∂V/∂y
= -2x. E_{z }= -∂V/∂z = 0.

The potential difference between the two plates of the capacitor shown below is
12 V. Equipotential surfaces are shown. If the separation between the
plates is 1 mm, what is the strength of the electric field between the plates?

Solution:

Let the y-axis point upward. The field is uniform and
points in the negative y-direction.

∆V/∆y = -E_{y}
= (12 V)/(10^{-3} m).

The strength of the electric field between the plates is E = 12000 V/m =
12000 N/C.

1 electron volt (eV) = 1.6*10^{-19 }J.

1 eV is the change in potential energy of a particle with charge q_{e} = 1.6*10^{-9}
C when the change in potential is 1 Volt (V). The unit was defined so that
when you know the voltage between two points in space, you know the change in
potential energy of an elementary particle when it moves from one to the other
point. (The sign of the change in potential energy depends on the sign of the
charge.) When a free proton moves through a potential difference of 1 V
its kinetic energy decreases by -qV = (1.6*10^{-19 }C)*(1 J/C) = -1.6*10^{-19 }J.
When a free electron moves through the same potential difference of 1 V its
kinetic energy increases by -qV = -(-1.6*10^{-19 }C)*(1 J/C) = 1.6*10^{-19 }J.