Physics

MOTION IN FIELDS   Electric Field and Electric Potential

Key Unit Competence

By the end of the unit, the learner should be able to analysis electric and potential fields.

My goals

By the end of this unit, I will be able to:

* define electric field and electric potential.

* explain the relationship between electric potential and electric field intensity.

* describe functioning of lightening arrestors.

* identify the dangers of lightening and how to avoid them.

Introduction

Have you ever heard sound due to lightening? If yes, what do you think was the cause?

If not, ask your friend in your class, at home, or neighbour about lightening.

Scientifically, lightening and thunder are effects of electric charges created in space (will be discussed later).

Attraction and repulsion of charges

Activity 1

In this section, you will observe the characteristics of the two types of charges, and verify experimentally that opposite charges attract and like charges repel.

Equipment

* Two lucite rods

* One rough plastic rod

* Stand with stirrup holder

* Silk cloth

* Cat’s fur

Procedure

1. Charge one lucite rod by rubbing it vigorously with silk. Place the rod into the stirrup holder as shown in Figure.

2. Rub the second lucite rod with silk, and bring it close to the first rod

3. . What happens? Record the observations in your notes.

4. Rub the rough plastic rod with cat’s fur, and bring this rod near the lucite rod in the stirrup. Record your observations.

5. What do you conclude?

6. Note down observation in your notebook.

For reference purposes, according to the convention originally chosen by Benjamin Franklin, the lucite rods rubbed with silk become positively charged, and the rough plastic rods rubbed with cat’s fur become negatively charged. Hard rubber rods, which are also commonly used, become negatively charged.

Coulomb’s law

Activity 2

Materials

* Coulomb’s Law apparatus

* Electrophorus (The electrophorus is a simple electrostatic induction device. It’s an inexhaustible source of charge”).

* Silk cloth.

* A computer for the graph and quick calculation.

Procedure

1. Take a moment to check to position of the hanging ball in your Coulomb apparatus. Look in through the side plastic window. The hanging ball should be at the same height as the sliding ball (i.e. the top of the mirrored scale should pass behind the centre of the hanging pith ball, as in Figure 10.3 below). Lift off the top cover and look down on the ball. The hanging ball should be centred on a line with the sliding balls. If necessary, adjust carefully the fine threads that hold the hanging ball to position it properly.

2. Charge the metal plate of the electrophorus in the usual way by rubbing the plastic base with silk, placing the metal plate on the base, and touching it with your finger.

3. Lift off the metal plate by its insulating handle, and touch it carefully to the ball on the left sliding block.

4. Slide the block into the Coulomb apparatus without touching the sides of the box with the ball. Slide the block in until it is close to the hanging ball. The hanging ball will be attracted by polarization, as in Section III of this lab. After it touches the sliding ball, the hanging ball will pick up half the charge and be repelled away. Repeat the procedure if necessary, pushing the sliding ball up until it touches the hanging ball.

5. Recharge the sliding ball so it produces the maximum force, and experiment with pushing it towards the hanging ball. The hanging ball should be repelled strongly.

6. You are going to measure the displacement of the hanging ball. You do not need to measure the position of its centre, but will record the position of its inside edge. Remove the sliding ball and record the equilibrium position of its inside edge that faces the sliding ball, which you will subtract from all the other measurements to determine the displacement d.

7. Put the sliding ball in, and make trial measurements of the inside edge of the sliding ball and the inside edge of the hanging ball. The difference between these two measurements, plus the diameter of one of the balls, is the distance r between their centres. Practice taking measurements and compare your readings with those of your lab partner until you are sure you can do them accurately. Try to estimate measurements to 0.2 mm.

8. Take measurements, and record the diameter of the balls (by sighting on the scale).

9. Remove the sliding ball, and recheck the equilibrium position of the inside edge of the hanging ball.

10. You can record and graph data in Excel or by hand (although if you work by hand, you will lose the opportunity for 2 mills of additional credit below). Recharge the balls as in steps 1 – 4, and record a series of measurements of the inside edges of the balls. Move the sliding ball in steps of 0.5 cm for each new measurement.

11. Compute columns of displacements d (position of the hanging ball minus the equilibrium position) and the separations r (difference between the two recorded measurements plus the diameter of one ball).

12. Plot (by hand or with Excel) d versus 1/r2. Is Coulomb’s Law verified?

13. For an additional credit of 2 mills, use Excel to fit a power-law curve to the data. What is the exponent of the r-dependence of the force? (Theoretically, it should be −2.000, but what does your curve fit produce)?

14. For your records, you may print out your Excel file with a table and graph of your numerical observations and any other electronic files you have generated.

Interpretation

Knowledge of the forces that exist between charged particles is necessary for an understanding of the structure of the atom and of matter. The magnitude of the forces between point charges was first investigated quantitatively in 1785 by Coulomb, a French scientist. The law he discovered is stated as follows:

“The force between two point charges is directly proportional to the product of charges divided by the square of their distance apart”.

Quick check

1. If two equal charges, each of 1C, are separated in air by a distance of 1 km, what would be the force between them?

2. Determine the force between two free electrons spaced 1AO apart.

3. Two equally charged pith balls are 3cm apart in air and repel each other with a force of 4x10−5N. Find the charge of each ball.4. How many electrons are contained in – 1C of charge? What is the total mass of these electrons?

Exercises

1. Two point charges q1 andq2 are 3m apart, and their combined charge is 20μC.

a) If one repels the other with a force of 0.075N, what are the two charges?

b) If the one attracts the other with a force of 0.525N, what are the magnitudes of charges?

2. Two point charges of q and q1 coulombs separated by a distant of 5m repel with a force of 0.072N. After having been put in contact, one replaces them at the same distance. Then they repel with a force of 0.081N. Calculate the charges before contact.

3. Two balls have identical masses of 0.1g each. When suspended to 10cm - long strings, they make an angle of 15° with the vertical. If the charges on each are the same, how large is each charge?

4. A test charge q = +2μC is placed halfway between a charge q = +6μC and a charge q = +4μC which are 10cm apart. Find the force on the charge test and its direction.

5. Three point charges are placed at the following points on the x axis: +2μC at x = 0, −3μC at x = 40cm, −5μC at x = 120cm. Find the force on the −3 μC charge.

6. Charges +2, +3 and −8μC are placed at the vertices of an equilateral triangle of side 10cm. Calculate the magnitude of the force acting on the −8μC charge due to the other charges.

Electric field

Notions and definitions

Questions to think about

a) You have learned about Coulomb’s law and you have seen that when an electric charge is brought near to another, there is an attractive or a repulsive force. Does that force acts when charges are in contact or it acts even at a certain distance?

b) If so, what can be the reason?

c) Does that force increase or decrease when the distance between charges increases?

After responding to those questions, you’ll see that around an electric charge is a region so that when another charge is placed in it, it undergoes an electric force. That region is called electric field created by the first charge.

An electric field can be defined as a region where an electric force is obtained. It’s a region where an electric charge experiences a force.

If a very small charge, positive point charge q is placed at any point in an electric field and it experiences a force F, then the field strength E (also called the E–field) at that point is defined by the equation: E= F/q.

In words, the magnitude of E is the force per unit charge and its direction is that of F(that is to say of the force which acts on a positive charge). Electric field is therefore a vector and we can write:

If F is in newtons [N] and q is in coulombs [C], then the unit of E is [NC^-1]. We shall see later that a more practical unit of E is volt–meter ^-1 [Vm-1].

E due to a point charge

The magnitude ofdue to an isolated positive point charge +q at the point P distance d away, in a medium of permittivity, ε, can be calculated by imagining a very small charge +q’ to be placed at P.

The following diagrams show it:

The above expression shows that E decreases with distance from the point charge according to an inverse square law. The field due to an isolated point charge is therefore non – uniform but it has same value at equal distances from the charge and so has spherical symmetry.

Examples

1. A charge, +6μC, experiences a force of 2mN in the +x direction at a certain point in space.

a) What was the electric field there before the charge was placed there?

b) Describe the force a −2μC charge would experience if it was used in place of +6μC

2. Find the electric field at a distance of 0.1m from a charge of 2ηC.3. Calculate the electric field created by a point charge of 1μC at a point situated at 2.5m in air and in water. The relative permittivity of water is 80.

Field lines (lines of force)

Activity 4: Lab zone Existence of field lines

This shows the shape of electric fields, in much the same way that magnetic fields are demonstrated with iron filings.

Materials

* Power supply, EHT, 0-5kV.

* Electric fields apparatus.

* Semolina.

* Castor oil.

Procedure

a) Fill the electrode unit with a layer of castor oil to a depth of about 0.5cm. Sprinkle a thin layer of semolina over the surface. (A thin piece of glass tubing drawn out to give a fine pointed stirrer is helpful so that the semolina is evenly distributed.) It is better to start with too little semolina than to start with too much. You can always increase the quantity later.

b) Place the electrodes in the castor oil. Connect the positive and negative terminals of the EHT power supply to the electrodes. Adjust the supply to give 3,000 to 4,000 volts. When the voltage is switched on, the field lines will be clearly visible.

c) Try electrodes of different shapes. For example, one can be a ‘point’ electrode whilst the other is a plate, or two point electrodes can be used. A wire circular electrode with a point electrode at the centre will show a radial field. The field with two plates quite close together should also be shown.

A line of force or field lines is defined as a line such that the tangent to it at a point is in the direction of force on a small positive charge placed the point.

Arrows on the lines of force show the direction of the force on a positive charge; the force on a negative charge.

Uniform electric field

A uniform electric field is one in whichhas the same magnitude and direction at all points, there is a plane symmetry and the field lines are parallel and evenly spaced.

This is the case for example of electric field between two parallel – plate carrying charges which have opposite signs.

Electric field due to a distribution of electric charges

Activity 5 Electric field due to a distribution of charges

Materials

* A sheet of paper

* A pen

*A ruler

Procedure

1. Represent a distribution of charges where you have charges of different signs.

2. Represent a point A where you want to find the total electric field.

3. At the point, A represents directions of electric fields vectors produced by each charge.

4. Do the sum of electric fields. Remember that an electric field is a vector. When they make a certain angle between them, use the method of parallelogram. When they have the same direction or opposite directions, use the appropriate method.

5. Establish a mathematical relation of the total electric field due to the distribution of charges.

The charge per unit area of the surface of the conductor is called the charge density σ (sigma) and since a sphere has the surface area 4πr²,we have; Therefore, q = 4π r²σ and so

This expression has been derived by considering a sphere but it gives E at surface of any charges conductor. It’s called Gauss' theorem.

Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple release of energy by objects to the realization of abilities in people.

Although the concept of electric potential is useful in understanding electrical phenomena, only differences in potential energy are measurable. If an electric field is defined as the force per unit charge, then by analogy an electric potential can be thought of as the potential energy per unit charge. Therefore, the work done in moving a unit charge from one point to another (e.g., within an electric circuit) is equal to the difference in potential energies at each point.

Potential difference, work, energy of charges

Activity 8

Potential energy, work, energy of charges

a) Consider two points A and B in an electrostatic field of strength E, and suppose that the force on a positive charge q has a component in the direction AB. Then if we move a positively charged body from B to A, we do work against this component of the field. The potential at A and at B are not equal. How can we define the potential difference between A and B?

b) From the definition in (a), if A_Vis the electric potential at the point A and B_V the electric potential at the point B. Knowing that if move a positive charge from A to B, the force produces a work W_AB. With which formula can we calculate the potential difference between A and B?

c) The unit of the potential difference has a special name called volt [V]. Can you find its unit in S.I units knowing that that 1V is equal to 1 unit of what you have to find?

d) Considering potential difference theory, the energy is expressed in another unit called electron-volt [V]. How can you define an electron-volt? What is the relation between an [eV] and a joule [J]?

e) From your knowledge, what is the instrument used to measure the potential difference in the circuit and how is it connected?

Electric potential is a location-dependent quantity that expresses the amount of potential energy per unit of charge at a specified location. When a Coulomb of charge (or any given amount of charge) possesses a relatively large quantity of potential energy at a given location, then that location is said to be a location of high electric potential. And similarly, if a Coulomb of charge (or any given amount of charge) possesses a relatively small quantity of potential energy at a given location, then that location is said to be a location of low electric potential. As we begin to apply our concepts of potential energy and electric potential to circuits, we will begin to refer to the difference in electric potential between two points.

Relation between E and V

Activity 9

Relation between E and V

1. What is the relation to find the work done by an electric force to move a charge from A to B, knowing that the distance between A and B is d?

2. What is the relation of the work using the potential difference?

3. Equalize the two relations and deduce the value of E. The relation found is the one between E and V.

4. From the expression found, deduce the new unit of the electric field E.

5. Write down the relation between E and V found, express in equation of V, write the electric field produced by a charge at a point deduce the electric potential created by a charge at a point situated at a distance d from it.

Quick check

1. Determine the work of an electric force which moves a point charge of q = 4μC from A to B if the p.d VA −VB =10V. Find the electric potential in vacuum at 0.2m from a charge of 2μC.

2. Between two parallel plates 1cm apart is a p.d of 200V. What energy is given to a charge of 1μC to move it from one plate to another? Calculate the value of the electric field between them.

3. Between two parallel plates 10cm apart is a electric field of 300V/m. Calculate:

(a) The voltage between them.

(b)The work of the electric force applied to an electron to move from one plate to another.

Exercises

1. Two spheres A and B charged negatively of radii 3cm and 9cm have an electric potential 300 000 V. Determine the distance between them so that the spheres repel with a force F = 0.3N.

2. Three equal charges of+6ηC are located at the corners of an equilateral triangle whose sides are 12cm length. Find the potential at the centre of the base of the triangle.

3. Suppose metal parallel plates are spaced 0.50cm apart and are connected to a battery. Find the electric field between them and the surface charge density on the plates.

4. The charge on an electron is 1.6×10−19 C in magnitude. An oil drop has a weight of 3.2×10¹³N. With an electric field of 5×10⁵ V/m between the plates of Millikan’s oil drop apparatus this drop is observed to be essentially balanced. What is the charge of the drop in electronic charge units?

5. In the Millikan experiment, an oil drop carries four electronic charges and has a mass of 1.8×10⁻¹² g. It is held almost at rest between two horizontal charged plates 1.8cm apart. What voltage must there be between the two charges plates?

6. Between two vertical parallel plates A and B exists a p.d V. The distance between the plates is 10cm. A small electrified ball of mass 0.3 g carrying a positive charge of 0.3μC is suspended to an insulating wire, of negligible mass that, the balance being realised forms an angle α=15° with the vertical. If g=9.8m/s².

(a) Calculate V.

(b) Trace some field lines between them, indicate their direction. (c)What work would be necessary to give to move the ball P and to bring it in the position P on the vertical of O; (length of the wire OP = l = 20cm).

Motion of electric charges in an electric field

Activity 10

a) Observe the picture and say it represents the inside of which apparatus.

b) You see a tube called cathode ray tube (CRT) Search on internet and give its main parts.

c) Doing the research, give a small idea about its principle of functioning.

In the process of functioning you’ll find that charges (electrons) are produced, are sent in motion in an electric field and reach a fluorescent screen. Here we are interested in the motion of the electric field.

On the figure below, charges, here we consider electrons, with a horizontal vector velocity of magnitude v₀ entering between two horizontal plates P₁ and P₂ separated by a distance

d. A p.d V = V_P1– V_P2 is applied between the plates.

We assume the electric field between the plates is uniform and acts on electrons on a horizontal distance l measured from 0. The point A is the point I where electrons get out the electric field; l is the distance through which the uniform field acts and x the horizontal trajectory travelled by electrons. In the electric field, an electric force acts vertically on the charges. So there is deflection of electrons in the electric field.

d) Why the upper plate must be charged positively and the lower plate charged negatively for this case?

e) If l = x, the motion being in the plane, find the equation of the horizontal motion.

f) Find the equation of the vertical motion.

g) Write down the second Newton’s law of the motion of those electrons, write the electric force from which electrons are subjected and deduce the acceleration of the motion.

h) Show that the trajectory of the motion between plates is a parabola and give its equation.

i) Calculate the velocity of electrons at the point A where they leave the electric field.

There are so many applications of cathode ray tube which is a practical example of the motion of electrons in an electric field in daily life. For example TV sets, oscilloscope, etc. use cathode ray tubes.

Lightening and lightening arrestor

Activity 11

Lightening and lightening arrestor

a) Surely, you have heard a thunder before the rainfall. What do you observe in the sky during it?

b) According to you, this is due to what?

c) Is the fact observed dangerous?

d) If yes, do you know some consequences which you have observed or heard?

e) If yes, is there a way to be protected from it?

f) Do some research on internet to know more about it and submit the result of your research to the teacher.

Some explanation

What you observe is called Lightening which is a sudden electrostatic discharge (the sudden flow of electricity between two electrically charged objects caused by contact, an electrical short, or dielectric breakdown) during an electrical storm between electrically charged regions of a cloud (called intra-cloud lightening or IC), between that cloud and another cloud (CC lightening), or between a cloud and the ground (CG lightening). The charged regions in the atmosphere temporarily equalise themselves through this discharge referred to as a strike if it hits an object on the ground. Although lightening is always accompanied by the sound of thunder, distant lightening may be seen but be too far away for the thunder to be heard. Lightening strikes can be damaging to buildings and equipment, as well as dangerous to people.

Buildings often use a lightening protection or lightening rod system consisting of a lightening rod (also called a lightening conductor) and metal cables to divert and conduct the electrical charges safely into the ground. Another form of lightening protection system creates a short circuit to prevent damage to equipment. The electrically conducting metal skin of commercial aircraft is isolated from the interior of to protect passengers and equipment.

Often, the lightening protection is mounted on top of an elevated structure, such as a building, a ship, or even a tree, electrically bonded using a wire or electrical conductor to interface with ground or “earth” through an electrode, engineered to protect the structure in the event of lightening strike. If lightening hits the structure, it will preferentially strike the rod and be conducted to the ground through the wire, instead of passing through the structure, where it could start a fire or cause electrocution. Lightening rods are also called finials, air terminals or strike termination devices.

In a lightening protection system, a lightening rod is a single component of the system. The lightening rod requires a connection to earth to perform its protective function. Lightening rods come in many different forms, including hollow, solid, pointed, rounded, flat strips or even bristle brush-like. The main attribute common to all lightening rods is that they are all made of conductive materials, such as copper and aluminum. Copper and its alloys are the most common materials used in lightening protection.

Exercise

1. Briefly describe how a lightening conductor can safeguard a tall building from being struck by lightening.

2. Find the force between two point charges +4μC and -3μC placed at a distance of 12dm apart in free space.

3. A charge of 4μC is placed in a vacuum. Determine the electric field intensity at a point P at a distance of 20cm from the charge.

4. The vertical deflecting plates in a Television set are 5.0cm and 1.0cm apart. If a potential difference of 100V is applied between the plates and the electron beam enters horizontally mid way between the plates with a speed of 2.0 × 10^-7ms^-1. Find the kinetic energy gained from the electric field by an electron in the beam.

5. The studied device is in an empty enclosure. The study is done with regard a mark supposed galilean; being horizontal and vertical. Electrons penetrate in O, with a horizontal velocity , inside a parallel plates capacitor. Between the parallel plates P₁ and P₂ of this capacitor separated by a distance d is applied a constant voltage V = V_P1-V_P2 = 140V. We’ll assume that the resulting electric field acts on electrons on a horizontal distance of 1m measured from O. Knowing that: Charge of an electron: e =1.6 x 10^-19 C; Mass of an electron:m= 9.1 x 10³¹kg ; Acceleration due to gravity: g = 9.8 m.s^-2; l = 15 cm; Velocity of electrons arriving in O: v_o = 30 000km s^-1; distance between the plates P₁ and P₂: d =3cm.

a) Compare the values of the weight of the electron and the electrostatic force undergone inside the capacitor. Conclude.

b) (i) Give the equations of coordinates x and y of the motion of the electron in the mark , when it passes between the plates P₁ and P₂.

(ii) Establish the equation of the trajectory of the electron.

c) With which vertical distance electrons are deviated at the exit of the capacitor?

(ii) What is the condition that electrons move out the electric field between the plates P₁ and P₂, the initial velocity keeping the above fixed value?

d) These electrons make a spot on a luminescent screen placed perpendicularly to and at a distance D = 20cm from the center C of the capacitor.

What is the distance of that spot to the center I of the screen?