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```TOPIC 5.1. ELECTRIC FIELDS
UNDERSTANDINGS:
• Charge
• Electric field
• Coulomb’s law
• Electric current
• Direct current (dc)
• Potential difference
APPLICATIONS AND SKILLS:
□ Identifying two forms of charge and the direction of the forces
between them
□ Solving problems involving electric fields and Coulomb’s law
□ Calculating work done in an electric field in both joules and
electronvolts
□ Identifying sign and nature of charge carriers in a metal
□ Identifying drift speed of charge carriers
□ Solving problems using the drift speed equation
□ Solving problems involving current, potential difference and charge
DATA BOOKLET:
NATURE OF SCIENCE
Modelling: Electrical theory demonstrates the scientific thought involved in
the development of a microscopic model from macroscopic observation.
The historical development and refinement of these scientific ideas when
the microscopic properties were unknown and unobservable is testament to
the deep thinking shown by the scientists of the time.
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ATOMS AND IONS
In a simplified atomic model, electrons orbit around a central nucleus. As long as the number of protons is equal
to the number of electrons, an atom is neutral.
If an electron is removed from an atom, the atom has a net (+) positive charge and become a positive ion.
If an electron is added from an atom, the atom has a net (-) negative charge and become a negative ion.
Hydrogen: neutral
Negative ion of
hydrogen
Beryllium: neutral
Positive ion of
Beryllium
2
ELECTRIC CHARGE
Electric charge is a property of matter. Ordinarily, matter appears electrically neutral but there are simple
experiments which put those charges in evidence.
Experiment:
- Take two plastic rods, rub each with a piece of wool and place the rods close to each other. Observe.
- Take two glass rods, rub each with silk and place the rods close to each other. Observe.
- Place the plastic rod and the glass rod close to each other. Observe.
Observation:
Interpretation:
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ELECTRIC CHARGE
▪ There are two types of charges: positive and negative.
▪ Like charges repel and unlike charges attract.
▪ The charge is CONSERVED: in a closed system, the amount of charge remains constant.
▪ Electric charge is QUANTISED. This means the amount of electric charge on a body is
always an integral multiple of a basic unit.
The basic unit is the magnitude of the charge on the proton and is equal to 1.6 × 10−19 C. This is called the
elementary charge, e.
Example 1
Two separated, identical conducting spheres are charged with charges of 4.0 μC and −12 μC, respectively. The
spheres are allowed to touch and then are separated again. Determine the charge on each sphere.
4
FREE ELECTRONS
In solid metals the atoms are fixed in position in a lattice but there are
many ‘free’ electrons that do not belong to a particular atom. These
electrons can move, carrying charge through the metal.
In liquids, and especially in gases, positive ions can also transport charge.
Materials that have many ‘free’ electrons are called conductors.
Materials that do not have many ‘free’ electrons, so charge cannot move freely,
are called insulators.
Thus, materials can be divided in 3 categories:
● Conductors: materials which allow the passage of
electric charge, due to the presence of free electrons.
● Non conductors (insulators): materials which do not
allow the passage of electric charge (no free electrons)
● Semi-conductors: lie between conductors and insulators
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THE ELECTROSCOPE
An electroscope is a simple device used to detect electrical charges. It was invented in 1600 and it is the first
electrical measuring instrument.
Structure of the electroscope:
-
An Erlenmeyer has a rubber stopper with a hole in it. Both rubber and glass are insulators
A conductor is passed through the hole
At the outside end of the conductor is a conductivity ball
At the inside end of the conductor is a very thin and flexible gold leaf than hangs under its
own weight. Gold is a good conductor.
Principle:
- A charge is placed on the ball and the conductors allow the charge to spread out, all the way to the
gold leaf.
- When the charge reached the leaf, since each leaf has some of the original charge, the leaves will repel
- When they repel, they spread out from one another.
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THE ELECTROSCOPE
Example 2
Consider the three electroscopes shown below.
a) Which one has the greatest charge?
b) Which one has the least charge?
c) Can you tell whether the charge is positive or negative? Why?
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COULOMB’S LAW
There is a force existing between two charged particles: the electric force.
This force is attractive for charges of opposite sign and repulsive for charges of
the same sign. It is defined as:
𝐹=𝑘
𝑞1𝑞2
1 𝑞1𝑞2
=
4πε0 𝑟²
𝑟²
F: electric force
r: distance between center of charges
q1, q2: point charges
k: Coulomb’s constant, k = 8.99 x 109 Nm2C-2
ε0: permittivity of vacuum, ε0 = 8.85 x 10-12 C²N-1m-2
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EXAMPLES
Example 3
a) Find the Coulomb force between two electrons located 1.0 cm apart.
b) If the two electrons are embedded in a chunk of quartz, having a permittivity of 120, what will the Coulomb
force be between them if they are 1.0 cm apart?
Example 4
A conducting sphere of radius 0.10 m holds an electric charge of Q = + 125 C. A charge q = -5.0 C is located
0.30 m from the surface of Q. Find the electric force between the two charges.
Example 5 (homework)
Two point charges of +10 nC and –10 nC in air are separated by a distance of 15 mm.
a) Calculate the force acting between the two charges.
b) Comment on whether this force can lift a small piece of paper about 2 mm × 2 mm in area.
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EXAMPLES
Example 6 (homework)
Two point charges of magnitude +5 μC and +3 μC are 1.5 m apart in a liquid that has a permittivity of 2.3 × 10–11
C2 N–1 m–2. Calculate the force between the point charges.
Example 7 (homework)
Two charges, q1 = 2.0 μC and q2 = 8.0 μC, are placed along a straight line separated by a distance of 3.0 cm.
a) Calculate the force exerted on each charge.
b) The charge q1 is increased to 4.0 μC. Determine the force on each charge now.
c) A positive charge q is placed on the line joining q1 and q2. Determine the distance from q1 where this third
positive charge experiences zero net force
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ELECTRIC FIELDS
The space around a charge or an arrangement of charges contains an electric field.
We can test whether a space has an electric field by bringing a small, point, positive charge q into the space.
That charge is called a test charge. If q experiences an electric force, then there is an electric field. If no force is
experienced, then there is no electric field
The electric field strength is defined as the electric force per unit charge experienced by a small, positive
point charge q:
𝐸=
𝐹
𝑞
11
EXAMPLES
Example 8
Calculate the electric field strengths in a vacuum
a) 1.5 cm from a +10 μC charge
b) 2.5 m from a –0.85 mC charge
Example 9
An oxygen nucleus has a charge of +8e. Calculate the electric field strength at a distance of 0.68 nm from the
nucleus
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PLOTTING ELECTRIC FIELDS
In order to explore the electric field surrounding a charge, we use that tiny POSITIVE test charge and we look at its
behaviour to determine the sign of the charge.
By placing a series of test charges around a negative/positive charge, we can map out its electric field.
Rules:
▪ Lines start and end on charges of opposite signs
▪ An arrow is essential to show the direction in which a positive charge would move
▪ Lines never cross
▪ Lines are closer together where the field is stronger
▪ Lines meet at conducting surface at 90°.
Examples:
Positive isolated charge
Negative isolated charge
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PLOTTING ELECTRIC FIELDS
* Field between parallel plates
* Field between two charges
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PLOTTING ELECTRIC FIELDS
* Field between a charge and a plate
Example 10
Which field is that of a) the largest negative charge b) the largest positive charge c) the smallest negative
charge d) the smallest positive charge
15
Electric fields can be added either by calculation or with a scale diagram to find the net electric field on a charge.
16
EXAMPLES
Example 11
Two charges of -0.225 C each are located at opposite corners of a square having a side length of 645 m.
Find the electric field vector at
(a) the center of the square, and
(b) one of the unoccupied corners.
Example 12
At which point is the electrical field the greatest?
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EXAMPLE
Example 13 (homework)
Two point charges, a +25 nC charge X and a +15 nC charge Y are separated by a distance of 0.5 m.
a) Calculate the resultant electric field strength at midpoint between the charges.
b) Calculate the distance from X at which the electric field strength is zero.
c) Calculate the magnitude of the electric field strength at the point P on the diagram. X and Y are 0.4 m and 0.3
m from P respectively.
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ELECTRIC FIELDS
* Field near the surface of a conductor
▪ Close enough to the conductor, the surface would appear flat
▪ Free electrons are equally spaced
▪ Field strength vectors parallel to the surface cancel out
On a sphere, the electric field is at 90° to the surface and is
The field outside a sphere behaves as if it came from a point
charge placed at the center of the sphere with a charge equal to
the total charge spread over the sphere.
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EXAMPLES
Example 14
An isolated metal sphere of radius 1.5 cm has a charge of -15 nC placed on it.
(a) Sketch in the electric field lines outside the sphere.
(b) Find the electric field strength at the surface of the sphere.
(c) An electron is placed on the outside surface of the sphere and released. What is its initial acceleration?
Example 15
If the charge on a 25 cm radius metal sphere is +150 C, calculate:
(a) the electric field strength at the surface.
(b) the field strength 25 cm from the surface.
(c) the force on a -0.75 C charge placed 25 cm from the surface.
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EXAMPLE
Example 16 (homework)
The uniform electric field strength inside the parallel plates is 275 N C-1. A +12 C
charge having a mass of 0.25 grams is placed in the field at A and released.
(a) What is the electric force acting on the charge?
(b) What is the weight of the charge?
(c) What is the acceleration of the charge?
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ELECTRIC CURRENT
In a conductor the ‘free’ electrons move randomly, much like gas molecules in
a container.
They do so with high speeds, of the order of 105 m s–1.
This random motion does not result in electric current; and the electric field inside a conductor is zero in static
situations.
If an electric field is applied across the conductor, the free electrons experience a force that pushes them in the
opposite direction to the direction of the field. This motion of electrons in the same direction is a direct current (dc)
We define electric current I in a conductor as the rate of flow of charge through its cross-section:
𝐼=
Δ𝑞
Δ𝑡
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DRIFT SPEED
The electric field inside a conductor follows the shape of the conductor.
Let’s consider electrons moving in a metallic wire. They move in the direction
opposite to the electric field with an average speed called the drift speed, v.
How many electrons will move through the cross-sectional area of
the wire within Δt?
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EXAMPLES
Example 17
Estimate the magnitude of the drift speed in a wire that carries a current of 1 A. The wire has radius 2 mm and
the number of electrons per unit volume (the number density) of free electrons is n = 1028 m−3 .
Example 18
Suppose the drift velocity is 0.0025 ms-1 for your house wiring.
a) If the wire between your light switch and your light bulb is 6.5 meters, how long does it take an electron to
travel from the switch to the bulb?
b) Why does the bulb still turn on instantly?
Example 19
State and explain whether the electron drift speed at B is
smaller than, equal to, or greater than that at A.
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EXAMPLES
Example 20 (homework)
In the shuttling ball experiment, the ball moves between the two charged plates at a frequency of 0.67 Hz.
The ball carries a charge of magnitude 72 nC each time it crosses from one plate to the other.
Calculate:
a) the average current in the circuit
b) the number of electrons transferred each time the ball touches one of the plates.
Example 21 (homework)
a) Calculate the current in a wire through which a charge of 25C passes in 1500 s.
b) The current in a wire is 36 mA. Calculate the charge that flows along the wire in one minute
Example 22 (homework)
A copper wire of diameter 0.65 mm carries a current of 0.25 A. There are 8.5 × 1028 charge carriers in each
cubic metre of copper; the charge on each charge carrier (electron) is 1.6 × 10–19 C. Calculate the drift speed of
the charge carriers.
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POTENTIAL DIFFERENCE
When charge q moves near other charges it will, in general, experience forces. So in moving the charge, work must
be done.
The potential difference is defined as the amount of work done per unit charge in moving a point charge from A
to B:
ΔV =
𝑊
𝑞
The actual path taken does not affect the amount of work that has to be done on the charge,
The joule is too large a unit of energy for the microscopic world. A more convenient unit (but not part of the SI
system) is the electronvolt, eV.
We define the electronvolt as the work done when a charge equal to one electron charge is taken across a
potential difference of one volt.
1 eV = 1.6 x 10-19 J
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EXAMPLES
Example 23
A charge of q = +15.0 C is moved from point A, having a voltage (potential) of 25.0 V to point B, having a
voltage (potential) of 18.0 V.
(a) What is the potential difference undergone by the charge?
(b) What is the work done in moving the charge from A to B?
Example 24
The work done in moving a charge of 2.0 μC between two points in an electric field is 1.50 × 10−4 J. Determine
the potential difference between the two points.
Example 25
An electron is moved from Point A, having a voltage (potential) of 25.0 V, to Point B, having a voltage (potential)
of 18.0 V.
(a) What is the work done (in eV and in J) on the electron by the external force during the displacement
(b) If the electron is released from Point B, what is its speed when it reaches Point A?
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EXAMPLES
Example 26
a) Determine the speed of a proton (m = 1.67 × 10–27 kg) that is accelerated from rest by a potential difference of
5.0 × 103 V.
b) A proton with speed 4.4 × 106 m s−1 enters a region of electric field directed in such a way that the proton is
slowed down. Determine the potential difference required to slow the proton down to half its initial speed.
Example 27 (homework)
A high efficiency LED lamp is lit for 2 hours. Calculate the energy transfer to the lamp when the pd across it is 240
V and the current in it is 50 mA.
Example 28 (homework)
A cell has a terminal voltage of 1.5 V and can deliver a charge of 460 C before it becomes discharged.
a) Calculate the maximum energy the cell can deliver.
b) The current in the cell never exceeds 5 mA. Estimate the lifetime of the cell.
28
EXAMPLE
Example 29 (homework)
The diagram shows two parallel plates situated in a vacuum.
One plate is at a positive potential with respect to the other.
A positively charged particle passes into the region between the plates.
Initially, the particle is travelling parallel to the plates.
(a) On the diagram,
(i) draw lined to represent the electric field between the plates
(ii) show the path of the charged particle as it passes between, and beyond, the plates.
(b) An electron is accelerated from rest in a vacuum through a potential difference of 750V.
(i) Determine the change in electric potential energy of the electron.
(ii) Deduce that the final speed of the electron is 1.6 x 107ms-1.
29
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