2nd Week

General Physics 2
Force of Electricity
IV. LESSON PROPER
4.1 COULOMB'S LAW
The mathematical formula to calculate the electrostatic force vector between two charged particles
is called Coulumb's law, named after the French physicist Charles Coulomb (1736-1806). Coulomb was the
𝑞 𝑞
first to propose a formula to calculate electrostatic force: 𝐹 = 𝑘 1𝑟 2 2 .
Coulomb's law calculates the magnitude of the force F between two point charges 𝒒𝟏 and 𝒒𝟐
separated by a distance r. In SI units, the constant k is equal to
𝑵. 𝒎𝟐 𝟗
𝒌 = 𝟖. 𝟗𝟗 𝒙 𝟏𝟎
𝑪𝟐
The electrostatic force (also called the Coulomb force) is defined as the amount and direction of
attraction or repulsion between two charged bodies.
Modern experiments have verified Coulomb's law to great precision. For example, it has been shown
1
that the force F is inversely proportional to distance between two objects squared (𝐹 ∝ 𝑟 2) to an accuracy
of 1 part in 1016 . No exceptions have ever been found even at the small distances within the atom.
The Coulomb force is a vector quantity and is expressed in units of newtons. The force is understood
to be along the line joining the two charges.
The magnitude of the electrostatic force F between point charges 𝒒𝟏 and 𝒒𝟐 , separated by a
distance r is given by Coulomb’s law. Note that Newton's third law (every force exerted creates an equal and
opposite force) applies as usual - the force on 𝒒𝟏 is equal in magnitude and opposite in direction to the force
exerts on 𝒒𝟐 , (a) Like charges (b) Unlike charges.
General Physics 2
4.2 THE ELECTRIC FIELD
Action at a distance is a force between objects that are not close enough for their atoms to “touch”,
meaning they are separated by more than a few atomic diameters. For example, a charged rubber comb
attracts neutral bits of paper from a distance via the Coulomb force.
Concept of a Field
A field is a way of conceptualizing and mapping the force that surrounds any object and acts on
another object at a distance without apparent physical connection. For example, the gravitational field
surrounding the earth (and all other masses) represents the gravitational force that would be experienced if
another mass were placed at a given point within the field.
In the same way, the Coulomb force field surrounding any charge extends throughout space. Using
|𝑞1 𝑞2 | |𝑄𝑞 |
Coulomb's law, = 𝑘 , its magnitude is given by the equation 𝐹 = 𝑘 or a point charge (a particle
𝑟2 𝑟2
having a charge Q) acting on a test charge q at a distance r. Both the magnitude and direction of the Coulomb
force field depend on Q and the test charge q.
To simplify things, we would prefer to have a field that depends only on Q and not on the test charge
q. The electric field is defined in such a manner that it represents only the charge creating it and is unique at
every point in space.
Specifically, the electric field E is defined to be the ratio of the Coulomb force to the test charge
𝐹
𝐸=
𝑞
where F is the electrostatic force (or Coulomb force) exerted on a positive test charge q. It is understood that
𝑬 is in the same direction as F. It is also assumed that q is so small that it does not alter the charge distribution
creating the electric field. The units of electric field are newtons per coulomb (N/C).
Electric Field Due to a Point Charge

Consider the electric field due to a point charge Q. According to Coulomb's law, the force in exerts
|𝑄𝑞 |
on a test charge q is 𝐹 = 𝑘 . Thus, the magnitude of the electric field E for a point charge
𝑟2
𝐹 𝑞𝑄 𝑄
𝐸 = |𝑞 | = 𝑘 |𝑞𝑟 2| = 𝑘 |𝑟 2 |.
𝑄
Since the rest charge cancels, we see that 𝐸 = 𝑘 𝑟 2 . The electric field is thus seen to depend only
on the charge Q and the distance r ; it is completely independent of the test charge q.
General Physics 2
The Coulomb force field due to a positive charge Q is shown acting on two different charges. Both
charges are the same distance from Q. (a) Since 𝑞1 is positive, the force 𝐹1 acting on it is repulsive. (b) The
charge 𝑞2 is negative and greater in magnitude than 𝑞1 , and so the force 𝐹2 acting on it is attractive and
stronger than 𝐹1 .
Example 1: Calculating the Electric Field of a Point of Charge

Calculate the strength and direction of the electric field due to a point charge of 2.00 nC (nanoCoulombs) at
a distance of 5.00 mm from the charge.
𝑄
Strategy: We can find the electric field created by a point charge by using the equation. 𝐸 = 𝑘 𝑟𝑞2
Solution: Here, 𝑄 = 2.00𝑥10−9 𝐶 and 𝑟 = 5.00𝑥10−3 𝑚. Entering those values into the above equation
gives.
𝑄𝑞
𝐸= 𝑘
𝑟2
𝑁. 𝑚2 2.0 𝑥 10−9 𝐶
𝐸 = 8.99 𝑥 109 𝑥
𝐶2 (5.00 𝑥 10−3 𝑚)2
𝐸 = 7.19 𝑥 105 𝑁/𝐶
Discussion: This electric field strength is the same at any point 5.00 mm away from the charge Q that creates
the field. It is positive, meaning that it has a direction pointing away from the charge Q.
Example 2: Calculating the Force Exerted on a Point Charge by an Electric Field

What force does the electric field found in Example 1.A exert on a point charge of −0.250𝜇𝐶 ?
Strategy: Since we know the electric field strength and the charge in the field, the force on that charge can
𝐹
be calculated using the definition of electric field 𝐸 = 𝑞 rearranged to 𝐹 = 𝑞𝐸.
General Physics 2
Solution: The magnitude of the force on a charge 𝑞 = −0.250𝜇𝐶 exerted by a field of strength
𝐸 = 7.19𝑥105 𝑁/𝐶 is thus,

𝐹 = 𝑞𝐸
= (−0.250𝑥10−6 𝐶) (7.19𝑥105 𝑁/𝐶)
= −0.180 𝑁
Discussion: Because q is negative, the force is directed opposite to the direction of the field. The force is
attractive, as expected for unlike charges. The charges in this example are typical of common static electricity,
and the modest attractive force obtained is similar to forces experienced in static cling and similar situations.
4.3 ELECTRIC FIELD LINES
Electric field lines are series of lines drawn from a point charge representing the magnitude and
direction of force exerted by that charge. Drawings using lines to represent electric fields around charged
objects are very useful in visualizing field strength and direction. Since the electric field has both magnitude
and direction, it is a vector. Like all vectors, the electric field can be represented by an arrow that has length
proportional to its magnitude and that points in the correct direction.
Two equivalent representations of the electric field to positive charge Q. (a) Arrow representing the
electric field's magnitude and direction (b) In the standard representation, the arrows are replaced by
continuous field lines having the same direction at any point as the electric field. The closeness of the lines
is directly related to the strength of the electric field. A test charge placed anywhere will feel a force in the
direction of the field line; this force will have a strength proportional to the density of the lines (being greater
near the charge, for example.)
Field lines are essentially a map of infinitesimal force vectors. Note that the electric field is defined
for a positive test charge q, so that the field lines point away from a positive charge and toward a negative
General Physics 2
charge. The electric field strength is exactly proportional to the number of field lines per unit area since the
𝑄
magnitude of the electric field for a point charge is 𝐸 = 𝑘 𝑟 2 and area is proportional to 𝑟 2 .
The electric field surrounding the different point charges. (a) A positive charge. b) A negative charge
of equal magnitude (c) A larger negative charge.
In many situations, there are multiple charges. The total electric field created by multiple charges is
the vector sum of the individual fields created by each charge.
While the electric fields from multiple charges are more complex than those of single charges, some
simple features are easily noticed. For example, the field is weaker between like charges, as shown by the
lines being farther apart in that region. (This is because the fields from each charge exert opposing forces on
any charge placed between them.) Furthermore, at a great distance from two like charges, the field becomes
identical to the field from a single, larger charge.
Properties of Electric Field Lines

Electric field lines are used to visualize and analyze electric fields. The properties of electric field
lines for any charge distribution can be summarized as follows:
 Field lines must begin on positive charges and terminate on negative charges, or at infinity in the
hypothetical case of isolated charges.
 The number of field lines leaving a positive change or entering a negative charge is proportional to
the magnitude of the charge.
 The strength of the field is proportional to the closeness of the field lines—more precisely, it is
proportional to the number of lines per unit area perpendicular to the lines.
 The direction of the electric field is tangent to the field line at any point in space.
 Field lines can never cross. This means that the field is unique at any point. The field line represents
the direction of the field; so if they crossed, the field would have two directions at that location (an
impossibility if the field is unique).
General Physics 2
Earth's Electric Field
Earth is an example of a near uniform electric field with approximately 150 N/C that is directed
downward, with the magnitude increasing slightly as it gets closer to the surface.
What causes the electric field? At around 100 km above the surface of Earth is a layer of charged
particles called the ionosphere. The ionosphere is responsible for a range of phenomena including the
electric field surrounding Earth. In fair weather, the ionosphere is positive and the Earth largely negative,
maintaining the electric field.
In storm conditions, clouds form and localized electric fields can be larger and reversed in direction.
The exact charge distributions depend on the local conditions, and variations are possible.
If the electric field is sufficiently large, the insulating properties of the surrounding material break
down and it becomes conducting. For air, this occurs at around 3 𝑥 106 N/C. Air ionizes ions and electrons
recombine, and we get discharge in the form of lightning sparks and corona discharge.

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General Physics 2

IV. LESSON PROPER

4.1 COULOMB'S LAW

4.2 THE ELECTRIC FIELD

Electric Field Due to a Point Charge

Example 1: Calculating the Electric Field of a Point of Charge

Example 2: Calculating the Force Exerted on a Point Charge by an Electric Field

𝐸 = 7.19𝑥105 𝑁/𝐶 is thus,

4.3 ELECTRIC FIELD LINES

Properties of Electric Field Lines

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