CHAPTER 2
Electrostatics in Conductors and Dielectrics
2.1. Conductors
We can generally define a conductor as a body that contains a large number of electric charge carriers that
are free to move. Now we need to be very specific about two terms: "electric charge carriers" and "free
to move". All bodies are made up of atoms and molecules. If any charged portion of any such atom
or a molecule inside the body is "free to move", then it satisfies the definition of a electric charge
carrier. "free to move" means that these particles have a high mobility, i.e. the particles can move a
reasonable distance within the body under the application of an electric field. A body that does not
satisfy this definition is called an insulator or dielectric. Do note that it is possible for an dielectric
to contain charge (and charged particles) and even store charge (and charged particles); however it
does not allow the charges to move about freely within itself. You can imagine them to be sort of
"frozen" inside the dielectric. Some common examples of dielectrics are: rubber, PVC plastic, cloth,
wood etc.
Conductors can be of many types. We’ll list some of the most common types below and some of their
characteristics:
• Metals - These are usually made up of a single element or an alloy of a mix of two or more
elements. Metals have a special property where one or more electrons from each atom are free
electrons. These free electrons can move anywhere within the metal; though they are loosely
bound to the lattice of atoms, they are not bound to one specific atom and can thus skip from
atom to atom. Here our electric charge carriers are the free electrons and their freedom of
motion gives them a high mobility. Common examples of metals are copper, iron, gold, silver
and aluminium
• Electrolytes - These are usually some salt or compound that is dissolved in a solvent like water.
The individual ions that make up the compound break apart in the solvent and form positive
and negatively charged ions that act as the electric charge carriers. Their freedom to move
in the solvent then gives them a high mobility. An example is sodium chloride solution (salt
water), where the positive sodium and the negative chlorine solutions are the charge carriers.
• Ionized gases/plasmas - Since the gases/plasma are already ionized, they become natural elec-
tric charge carriers. In addition, gases have a very low density which allows the carriers to have
a high mobility without being impeded by other gas molecules. Any sort of ionized gas will be
a valid example for this type of conductor.
50
,CHAPTER 2. ELECTROSTATICS IN CONDUCTORS AND DIELECTRICS 51
We also have a special class of materials called semiconductors which are an intermediate material
between conductors and dielectrics. These materials normally tend to behave as insulators, but on
"doping" with a suitable impurity, they become conductors if a minimum electric field is applied.
We will not discuss them further in this course, but will encounter them in detail in later electronics
courses as they form the backbone of that field of study.
2.1.1 Properties of conductors in equilibrium
We will discuss the case when the conductor is in electrostatic equilibrium, i.e. none of the electric
charges within the conductor is moving.
1. The electric field inside a conductor is zero (⃗E = 0)
Let us take the case of metals as our example of a conductor. Let’s say the conductor is inside
some external electric field ⃗Eext , as seen in Figure1 2.1.1. This field will cause all the free elec-
trons to move to one end of the conductor, making that region negatively charged (see Figure2
2.1.2). Any free positive charges will move in the opposite direction, making that region pos-
itively charged3 . Since we have a charge difference across the two ends of the conductor, it
will set up an electric field inside the conductor (⃗Eint ) that is opposite in direction to ⃗Eext . The
charges will continue to move until ⃗Eext = ⃗Eint , i.e. the two fields cancel each other out leaving
the total electric field inside the conductor equal to zero.
In the case when the conductor is not within any external electric field, the negative (or positive)
charge carriers will automatically repel each other and they will move as far as they possible
can away from each other. Obviously this would be at the surface of the conductor, leaving the
interior empty of charges. This again leaves the electric field inside the conductor as zero.
F IGURE 2.1.2: Movement of charges due to an ex-
F IGURE 2.1.1: Conductor in an external electric field ternal electric field
2. The charge density inside a conductor is zero (ρ = 0)
This should be obvious from Gauss’ law. We had ∇ • ⃗E = ρϵenc 0
. So if ⃗E = 0, it automatically
means that ρ = 0. Note that this means that the total charge density is zero, not that charge
itself does not exist inside the conductor. It just means that if there is any negative charge
somewhere in the conductor, there will be an equal positive charge somewhere else making the
overall charge zero.
3. All charges within a conductor will be located at it surface.
This should be obvious from our discussion of the first two points.
1
Image taken from Bettini, A. (2016). A course in classical physics 3 - Electromagnetism. Springer. Page 128.
2
Ibid.
3
Even if we consider the case of atoms in a metal, where the positive charges cannot physically move, the free elec-
trons will all move to one end. This means that there are less negative charges on the opposite end, making that region
automatically positively charged just from the motion of electrons alone.
, CHAPTER 2. ELECTROSTATICS IN CONDUCTORS AND DIELECTRICS 52
4. The conductor is an equipotential volume.
An equipotential volume means that the electric potential anywhere within the volume of
the conductor will have the same constant value (hence the term equipotential). This is also
simple to show. Take any two points a and b inside or on the surface of the conductor. We have
´b
V(b) − V(a) = − a ⃗E • d⃗l. Since ⃗E = 0, this implies that V(b) − V(a) = 0 =⇒ V(a) = V(b).
Thus the conductor is equipotential everywhere within itself.
5. The electric field just outside the conductor will be directed perpendicular to the surface.
This comes from our understanding of the boundary conditions discussed in the last chapter.
The surface of the conductor is an electrostatic boundary. Within the conductor ⃗E = 0 which
means both the normal and tangential components are zero. We know that the value of the
tangential component does not change across a boundary, so the tangential component of the
electric field outside will also be zero. We also know that the normal component will have
discontinuity of σ/ε0 , where σ is the surface charge density. Thus the only component that
exists outside the conductor is the normal component, which is perpendicular to the surface.
2.1.2 Induced charges and conductors
F IGURE 2.1.3: Free charge near a neutral conductor F IGURE 2.1.4: Charge inside a hollow conductor
Let a free positive charge q be brought near an isolated, neutral conductor as seen in Figure4 2.1.3.
The free charge will induce the negative charges in the conductor to move near it due to Coulomb
attraction, which pushes all the positive charges to the opposite end. Since there are negative charges
near the positive free charge, there will be an electrostatic force of attraction between the free charge
and the conductor (and we will see the charge try to move towards the conductor due to this reason).
The electric field inside the conductor will still be zero, however.
Now let us look at the case of the hollow conductor, as seen in Figure5 2.1.4. There is a positive
charge q inside the hollow region. Due to the charge, the electric field in the hollow region is not
zero. This charge will induce an equal charge of −q on the inner surface of the hollow. If we look at
the Gaussian surface drawn, the total enclosed charge is still zero (+q from the point charge and −q
from the induced charge) so the electric field within the conductor inside the Gaussian surface is still
zero. However the induced −q charge will push all the positive charges within the conductor to the
surface until they are also equal to +q. From outside the conductor we thus see a total charge of +q
distributed all over its outer surface, which tells us that there is a charge of +q located somewhere
inside the conductor. This way a conductor can give information about any free charge within itself.
4
Image taken from Griffiths, D.J. (2013). Introduction to electrodynamics (4th ed.). Pearson. Page 100.
5
Ibid.
Electrostatics in Conductors and Dielectrics
2.1. Conductors
We can generally define a conductor as a body that contains a large number of electric charge carriers that
are free to move. Now we need to be very specific about two terms: "electric charge carriers" and "free
to move". All bodies are made up of atoms and molecules. If any charged portion of any such atom
or a molecule inside the body is "free to move", then it satisfies the definition of a electric charge
carrier. "free to move" means that these particles have a high mobility, i.e. the particles can move a
reasonable distance within the body under the application of an electric field. A body that does not
satisfy this definition is called an insulator or dielectric. Do note that it is possible for an dielectric
to contain charge (and charged particles) and even store charge (and charged particles); however it
does not allow the charges to move about freely within itself. You can imagine them to be sort of
"frozen" inside the dielectric. Some common examples of dielectrics are: rubber, PVC plastic, cloth,
wood etc.
Conductors can be of many types. We’ll list some of the most common types below and some of their
characteristics:
• Metals - These are usually made up of a single element or an alloy of a mix of two or more
elements. Metals have a special property where one or more electrons from each atom are free
electrons. These free electrons can move anywhere within the metal; though they are loosely
bound to the lattice of atoms, they are not bound to one specific atom and can thus skip from
atom to atom. Here our electric charge carriers are the free electrons and their freedom of
motion gives them a high mobility. Common examples of metals are copper, iron, gold, silver
and aluminium
• Electrolytes - These are usually some salt or compound that is dissolved in a solvent like water.
The individual ions that make up the compound break apart in the solvent and form positive
and negatively charged ions that act as the electric charge carriers. Their freedom to move
in the solvent then gives them a high mobility. An example is sodium chloride solution (salt
water), where the positive sodium and the negative chlorine solutions are the charge carriers.
• Ionized gases/plasmas - Since the gases/plasma are already ionized, they become natural elec-
tric charge carriers. In addition, gases have a very low density which allows the carriers to have
a high mobility without being impeded by other gas molecules. Any sort of ionized gas will be
a valid example for this type of conductor.
50
,CHAPTER 2. ELECTROSTATICS IN CONDUCTORS AND DIELECTRICS 51
We also have a special class of materials called semiconductors which are an intermediate material
between conductors and dielectrics. These materials normally tend to behave as insulators, but on
"doping" with a suitable impurity, they become conductors if a minimum electric field is applied.
We will not discuss them further in this course, but will encounter them in detail in later electronics
courses as they form the backbone of that field of study.
2.1.1 Properties of conductors in equilibrium
We will discuss the case when the conductor is in electrostatic equilibrium, i.e. none of the electric
charges within the conductor is moving.
1. The electric field inside a conductor is zero (⃗E = 0)
Let us take the case of metals as our example of a conductor. Let’s say the conductor is inside
some external electric field ⃗Eext , as seen in Figure1 2.1.1. This field will cause all the free elec-
trons to move to one end of the conductor, making that region negatively charged (see Figure2
2.1.2). Any free positive charges will move in the opposite direction, making that region pos-
itively charged3 . Since we have a charge difference across the two ends of the conductor, it
will set up an electric field inside the conductor (⃗Eint ) that is opposite in direction to ⃗Eext . The
charges will continue to move until ⃗Eext = ⃗Eint , i.e. the two fields cancel each other out leaving
the total electric field inside the conductor equal to zero.
In the case when the conductor is not within any external electric field, the negative (or positive)
charge carriers will automatically repel each other and they will move as far as they possible
can away from each other. Obviously this would be at the surface of the conductor, leaving the
interior empty of charges. This again leaves the electric field inside the conductor as zero.
F IGURE 2.1.2: Movement of charges due to an ex-
F IGURE 2.1.1: Conductor in an external electric field ternal electric field
2. The charge density inside a conductor is zero (ρ = 0)
This should be obvious from Gauss’ law. We had ∇ • ⃗E = ρϵenc 0
. So if ⃗E = 0, it automatically
means that ρ = 0. Note that this means that the total charge density is zero, not that charge
itself does not exist inside the conductor. It just means that if there is any negative charge
somewhere in the conductor, there will be an equal positive charge somewhere else making the
overall charge zero.
3. All charges within a conductor will be located at it surface.
This should be obvious from our discussion of the first two points.
1
Image taken from Bettini, A. (2016). A course in classical physics 3 - Electromagnetism. Springer. Page 128.
2
Ibid.
3
Even if we consider the case of atoms in a metal, where the positive charges cannot physically move, the free elec-
trons will all move to one end. This means that there are less negative charges on the opposite end, making that region
automatically positively charged just from the motion of electrons alone.
, CHAPTER 2. ELECTROSTATICS IN CONDUCTORS AND DIELECTRICS 52
4. The conductor is an equipotential volume.
An equipotential volume means that the electric potential anywhere within the volume of
the conductor will have the same constant value (hence the term equipotential). This is also
simple to show. Take any two points a and b inside or on the surface of the conductor. We have
´b
V(b) − V(a) = − a ⃗E • d⃗l. Since ⃗E = 0, this implies that V(b) − V(a) = 0 =⇒ V(a) = V(b).
Thus the conductor is equipotential everywhere within itself.
5. The electric field just outside the conductor will be directed perpendicular to the surface.
This comes from our understanding of the boundary conditions discussed in the last chapter.
The surface of the conductor is an electrostatic boundary. Within the conductor ⃗E = 0 which
means both the normal and tangential components are zero. We know that the value of the
tangential component does not change across a boundary, so the tangential component of the
electric field outside will also be zero. We also know that the normal component will have
discontinuity of σ/ε0 , where σ is the surface charge density. Thus the only component that
exists outside the conductor is the normal component, which is perpendicular to the surface.
2.1.2 Induced charges and conductors
F IGURE 2.1.3: Free charge near a neutral conductor F IGURE 2.1.4: Charge inside a hollow conductor
Let a free positive charge q be brought near an isolated, neutral conductor as seen in Figure4 2.1.3.
The free charge will induce the negative charges in the conductor to move near it due to Coulomb
attraction, which pushes all the positive charges to the opposite end. Since there are negative charges
near the positive free charge, there will be an electrostatic force of attraction between the free charge
and the conductor (and we will see the charge try to move towards the conductor due to this reason).
The electric field inside the conductor will still be zero, however.
Now let us look at the case of the hollow conductor, as seen in Figure5 2.1.4. There is a positive
charge q inside the hollow region. Due to the charge, the electric field in the hollow region is not
zero. This charge will induce an equal charge of −q on the inner surface of the hollow. If we look at
the Gaussian surface drawn, the total enclosed charge is still zero (+q from the point charge and −q
from the induced charge) so the electric field within the conductor inside the Gaussian surface is still
zero. However the induced −q charge will push all the positive charges within the conductor to the
surface until they are also equal to +q. From outside the conductor we thus see a total charge of +q
distributed all over its outer surface, which tells us that there is a charge of +q located somewhere
inside the conductor. This way a conductor can give information about any free charge within itself.
4
Image taken from Griffiths, D.J. (2013). Introduction to electrodynamics (4th ed.). Pearson. Page 100.
5
Ibid.
