The table of tangents holds values for every acute angle from 0o to 90o. Each angle has a unique tangent ratio. We get this ratio when two lines meet to make the angle.
Every combination of opposite and adjacent lines that makes a right angled triangle has a unique angle which they make.
If we know the acute angle in a right angled triangle, we can use tables of tangents. This helps us find its corresponding tangent ratio. Similarly, if we know the angle and just one side, we can find the angle’s ratio. Then, we use the tangent relationship to find the other side.
The table of tangents consists of angles from 0o to 90o. We express these angles in 4 significant figures and record their values in a table. All we need to do as mathematician is get a certain angle and find it’s corresponding ratio from the tables.
We expresses angles in the table of tangents in degrees, points of degrees and as well as in minutes. 1 degree (1o) is equivalent to 60 minutes(60′).
We have divided the table of tangents into three major columns as shown in the table extract below:
The first column represents whole number degrees from 0o to 90o and has column head labeled xo which represents
The second column consists of 0.0o to 0.9o which divides a degree into 10 smaller units hence giving an accuracy of 0.1o.
The third column is the one we have labeled ADD and it provides the second decimal value of the angle. Using the table of tangents, we can find angles u to second decimal places.
Example
Determine the tangent of 36.57o
solution
In the column labelled xo , look for the row headed 36 and then move along this row until you reach 0.5. The number at the intersection of 36 and 0.5 is 0.7400
note that the number is recorded as 7400 and not 0.7400. This is done to save on space but you should check the first column after 36, That is, column headed 0.0, whatever value that is stated on that row in that column should be used as the starting value for all the columns in that row.
so tan 36.5 =0.7400, to get the value for tan 36.57, we go to the add column and check on the column 0.07 and add it’s value on the far right of our previous value we read from the table. In this case it is 19 and should be read as 0.0019
hence tan 36.57 should be 0.7400+0.0019 = 0.7419
Example
Use tables to find the tangent of 77o48′
solution
1o=60′, hence 48′ = (48′ x 1o)/60′ = 0.8o
then 77o48′ can be expressed as 77.8o
From the tables, you identify row 77 at xo column then move up to to the column 0.8 and read off that value at the intersection. This value is 0.6252 hence tan 77o48′ = tan 77.8o = 0.6252
Here are exam questions on waves that are common in national exams.
State two differences between electromagnetic waves and mechanical waves (2 marks)
Figure 3 show straight waves incident on a divergent lens placed in a ripple tank to reduce its depth.
Complete the diagram to show the waves in both the shallow region and beyond the lens (2 marks)
3. A ship in an ocean sends out an ultra sound whose echo is received after 3 seconds. if the wavelength of the ultra sound in water is 7.5 cm and the frequency of the transmitter is 20 kHz, determine the depth of the ocean. (3 marks)
4. Explain the fact that radiant heat from the sun penetrates a glass sheet while radian heat from burning wood is cut off by the glass sheet. (2 marks)
Question 5
5. (a) figure 5 shows a displacement-time graph for a progressive wave.
figure 5
(i) State the amplitude of the wave (1 mark)
(ii)Determine the frequency of the wave (4 marks)
(iii) Given that the velocity of the wave is 20 ms-1 , determine it’s wavelength. (3 marks)
(b)Figure 6 shows two identical dippers A and B vibrating in water in phase with each other . The dippers have the same constant frequency and amplitude. The waves produced are observed along the line MN:
Figure 6
It is observed that the amplitude are maximum at points Q and S and minimum at points P and R.
(i) Explain why the amplitude is maximum at Q. (2 marks)
(ii) state why the amplitude is minimum at R (1 mark)
(iii) State what would have happen if the two dippers had different frequencies . ( 1 mark)
6. Figure 7 shows water waves incident on a shallow region of the shape shown with dotted line.
Figure 7
On the same diagram, sketch the wave pattern in and beyond the shallow region (1 mark)
7 . Figure 7 shows standing wave on a string. It is drawn to a scale of 1:5
Figure 7
(a) Indicate on the diagram the wavelength of the standing wave (1 mark)
(b) Determine the wavelength of the wave. (1 mark)
Figure 15 shows two coils A and B placed close to each other. A is connected to a steady D.C supply and a switch, B is connected to a sensitive galvanometer.
Figure 15
(i) The switch is closed . State the observations made on the galvanometer (2 marks)
(ii) Explain what would be observed if the switch is then opened. (2 marks).
2. The primary coil of a transformer has 1000 turns and the secondary coil has 200 turns. The primary coil is connected to a 240V a.c mains, supply.
(i) Explain how an e.m.f is induced in the secondary coil (2 marks)
(ii) Determine the secondary voltage (3 marks)
(iii) Determine the efficiency of the transformer given that the current in the primary coil is 0.20 A and in the secondary coil it is 0.80 A
Question 3
3 .(a)(I) State Faraday’s law of electromagnetic induction (1 mark)
(ii) State the Lenz’s law of electromagnetic induction (1 mark)
(b) A transformer is used on a 240 a.c supply to deliver a 7.5 A at 90 V to a heating coil. if the transformer is 95% efficient, what is the current in the primary winding? (2 marks).
( c) Hysteresis losses are a source of inefficiency in a transformer. explain
(I) What is meant by hysteresis losses (2 marks)
(ii) How these hysteresis losses can be minimized. (1 mark)
( d) An a.c supply lights a lamp with the same brightness as a 12 V battery. Determine
(i)The r.m.s voltage (1 mark)
(ii) The peak voltage of the a.c supply (2 marks)
Question 4
4. (a) State the lenz’s law (1 mark)
(b) The diagram in figure 7 below shows a coil of wire next to a magnet. A voltmeter is connected to the coil of the wire.
Describe two ways of inducing a voltage in the coil of the wire. (2 marks)
(c) State the three factors that affects the size of the induced voltage (3 marks)
(d) A 6v, 24w lamp shines at a full brightness when it is connected to the output of a mains transformer as shown in figure 8 below.
Assuming that the transformer is ideal, calculate
(i) The number of turns in the secondary coil if the lamp is to work at it’s normal brightness (2 marks)
(ii) The current which flows in the mains cables (2 marks)
Question 5
5. Figure 14 shows an E shaped steel block being magnetised by a current through two coils in series.
on the figure, indicate
(i) the north and south poles of the resulting magnet (1 mark)
(ii) the complete magnetic field pattern between the poles (1 mark)
(b) Figure 15 shows the permanent magnet made in part (a) above
Figure 15
A coil wound loosely on the middle limb is connected in series with low voltage a.c and a switch. State and explain the observations made on the coil when the switch is closed (2 marks)
Question 6
6.A piece of metal AB was magnetised using the method shown below.
(a) By what method is metal AB being magnetised? (1 mark)
(b) what is the polarity of end B (1 mark)
Question 7
7. Figure 2 shows a soft iron bar AB placed in a coil near a freely suspended magnet.
Explain the observation made when the switch is closed (2 marks)
Question 8
8. When a transformer is connected to an ac source, the output voltage is found to be 24 V. If the power input is 200 W, determine the output current. (assume the transformer is 100% efficient). (3 marks)
9. (a) State what is meant by the term “electromagnetic induction”. (1 mark).
9 (b) Figure 9, shows a simple electric generator.
(i) Name the parts labelled P and Q. (2 marks)
P ……………………………………………
Q …………………………………………..
(ii) Sketch on the axes provided a graph to show how the magnitude of the potential difference across R, changes with time t. (1 mark)
(iii) State two ways in which the potential difference produced by such a generator can be increased. (2 marks)
(c) In a transformer, the ratio of primary turns to the secondary turns is 1:10. A current of 500mA flows through a 200 ohms resistor in the secondary circuit. Assuming that the transformer is 100% efficient, determine:
Mutual induction is when current is induced in a coil due to changing magnetic flux of a second coil nearby. The changing magnetic flux in the first coil links with the second coil inducing an e.m.f in it.
Demonstrating mutual induction
The figure below consists of two coils P and S , battery, a. c power source, rheostat, galvanometer and switch connected as shown.
When we close the switch K, the point on the galvanometer in the second coil deflects in one direction. It then returns to zero. When we open the switch, the pointer deflects to the opposite direction before falling back to zero. when we increase the primary current, more deflection is noted in either direction. If we replace a d.c source with an a.c power source , the pointer vibrates about point zero.
Explaining the observations on the demo
Current Strength:
5
Ready for demonstration.
When we closed the switch, current in the primary coil increases from zero to maximum within a short time. As the current builds up, magnetic flux in the primary coil increases from zero to its maximum value. This change links with the secondary coil inducing an e.m.f in the secondary coil.
Deflection of the galvanometer indicates that current is flowing in the secondary coil where it is connected. The current flow is momentarily. It flows once and stops. This is because the induced e.m.f in the secondary coil is momentarily . It happens when the current in the primary coil is building up from zero to maximum. When the current reaches the maximum value, the magnetic flux stops changing in the primary coil. Therefore, there is no further induction in the primary coil.
Opening the switch causes the primary current to decay from maximum to zero. This happens in a very short time. Consecutively, magnetic flux in the primary coil linking with the secondary coil falls from maximum to zero. This causes changing magnetic flux in the primary coil that induces an e.m.f in the secondary coil.
Induced e.m.f due to current decay
We observes more deflection when switch is opened than when the switch is being closed. This shows that the induced e.m.f in the secondary coil is much higher when switch is opened than when it is closed. The reason behind this is that current in the circuit takes much shorter time to decay than to build up.
Increasing current in the primary coil causes deflection on the galvanometer. This indicates that increasing current in primary coil causes changing magnetic flux in that coil hence continued induction of e.m.f in the secondary coil.
Continuously decreasing current in the primary coil causes an e.m.f in the secondary coil due to decreasing magnetic flux linking with the secondary coil.
From the Lenz’s law; the direction of the induced current when current is building up is as indicated in the diagram below:
when the current is decaying, the direction of current according to Len’z law is as shown.
Increasing efficiency in mutual induction
The induced e.m.f in the secondary coil can be increased by winding the primary and secondary coils on a soft iron rod. The soft iron concentrates magnetic flux in both coils. See the figure below.
A more efficient way of linking magnetic flux between the two coils is by winding the two coils on a soft iron ring as shown.
More induced e.m.f can be attained by having more turns in the secondary coil. The e.m.f is induced in each turn of the secondary coil. This occurs because the magnetic flux of the primary coil links with each of the turns. The total e.m.f is thus the summation of the individual e.m.f from each turn.
The Grade 9 to Senior School selection process in Kenya involves students choosing their preferred pathways. They also choose subject combinations and schools. This is done through an online system managed by the Ministry of Education.
This process is part of the transition to Senior School under the Competency-Based Education (CBE) framework. Students will select their pathways and subject combinations and they will also choose up to 12 schools across four clusters. STEM is a mandatory pathway.
SELECTION OF PATHWAYS AND SENIOR SCHOOLS
Determination of pathways per senior school
Determination of vacancies for boarding and day schooling in senior schools
Selection of pathways, subjects’ combination and schools by grade 9 learners
Selection based on pathway
The learner will select 12 schools for their chosen pathway as follows.
4 schools in first choice track and subject combination
Four (4) schools in second choice subject combination
Four (4) schools in third choice subject combination (Total 12 schools)
Selection based on accommodation
Out of the 12 schools selected based on pathway:
9 will be boarding schools; 3 from the learners’ home county, 6 from outside their home county/county of residence.
Three (3) day schools in their home sub county/sub county of residence. (Total 12 schools) Pre selection – A school that does not allow open placement can apply to be pre-select if it meets the criteria defined by the Ministry of Education.
Accommodation- Based Breakdown
Top 6 learners per gender in each STEM track per sub-county will be placed for Boarding in schools of choice
Top 3 learners per gender in each Social Science track per sub-county will be placed for Boarding in schools of choice
Top 2 learners per gender in each Arts and Sports Science track per sub-county be placed to Boarding schools of their choice
Placement of Candidates with Achievement Level of averaging 7 and 8 per track to boarding schools of their choice
The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration.
It essentially states that these two operations are inverses of each other, and it provides a way to evaluate definite integrals using anti-derivatives.
Suppose that f is continuous at a closed interval [a, b] . If the function F is defined on a closed interval [a, b] by:
$$F(x) = \int_{a}^{x} f(t) dt $$
where a is a real number, Then F is the anti-derivative of f. in other words, F'(x) = f(x)
consider the relationships:
then
f(x) = x2and
Note: We use the dummy variable (t) in the integrand to avoid confusion with the upper limit x.
Sometimes the fundamental theorem of calculus is interpreted to mean that:
differentiation and integration are inverse processes to each other.
It follows that:
The fundamental theorem of calculus states that:
if f is continous on an open interval containing a and x and then we first integrate the function f and then differentiate with respect to x, then the result we get is the function f again.
In other words, the fundamental theorem of calculus argues that differentiation cancels the effect of intergration of continous f(x’).
in short:
For example
Example problem1
Use the fundamental theorem of calculus to find derivative of the following functions
(a)
solution
NOTE: The best way to benefit from this examples is trying the problem first before looking for answers and attempting again after checking your work against the answer.
Example problem2
(b)
solution to problem 2
Example problem 3
Find h'(x) given that :
solution
let y=h(x) and u=x2 and hence:
since u=x2;
and therefore:
By use of chain rule:
which implies u3sinu(2x) = (x2)3sin(x2)2x resulting to:
=2x7sin(x2)
Example problem 4
Consider the expression below, we exchange the limits in the intergral and then change the sign from positive to negative before using the fundamental theorem to solve it.
Example problems on fundamental theorem of calculus
We exchange limits and so the sign of the integral so that the upper limit is the valuable x.
Example problem 6
Use the fundamental theorem of calculus to solve:
Solution
splitting the integral about point zero we have:
and then exchanging limits in the first integral;
let u=-x; first part of the expression above becomes;
from laws of differentiation du/dx=-1 and using chain rule;
The most common calculator used by high school learners is ms fx-82 model. Calculator has many scientific functions that can help a student work out many scientific computations. However, ms fx-82 calculator is non-programmable.
Casio fx-82MS Scientific Calculator
The Casio fx-82 MS is a widely used scientific calculator. It is especially favored by students and professionals for its reliability. It also complies with exam requirements.
Key Features of fx-82 calculator:
240 Functions: Includes trigonometric, statistical, fractional, and exponential calculations.
Natural Textbook Display: Shows expressions as they appear in textbooks for easy comprehension.
Two-Line Display: Simultaneously displays input and output for clarity.
Multi-Replay Function: Allows quick recall and editing of previous formulas.
STAT-Data Editor: Supports mean, standard deviation, and regression analysis.
9 Variable Memories: Stores and recalls up to 9 data sets for efficiency.
Durable Design: Features robust plastic keys and a protective slide-on hard case.
Battery Powered: Operates on a single AAA battery.
Non-Programmable: Compliant with exam standards for academic use.
Portable and Lightweight: Compact design for easy everyday use.
Handling Precautions
Even if the calculator is operating normally, replace the battery at least after every two years. Continued use after the specified number of years can result to abnormal operation.
You should replace the battery immediately after display figures become dim.
A dead battery can leak, causing damage to and malfunction of the calculator. Never leave a dead battery in the calculator.
The battery that comes with the calculator is for factory testing, and it discharges slightly during shipment and storage. Because of these reasons, its battery life can be shorter than normal. For that reason, consider replacing the battery sooner.
Avoid use and storage of the calculator in areas subjected to temperature extremes, and large amounts of humidity and dust.
Do not subject the calculator to excessive impact, pressure, or bending.
Never try to take the calculator apart.
Use a soft, dry cloth to clean the exterior of the calculator
Do not use a nickel-based primary battery with this product.
use of incompatible batteries such as nickel-based primary battery with fx-82ms calculator can result in shorter battery life and product malfunctioning.
press 2 and the use > and < to adjust display contrast. when satisfied with the display settings, press AC button
use of calculator keys
To use the alternate function of a key, press [SHIFT] key followed by the key. The alternate function is indicated by the text printed above the key. The alternate function is usually marked with a different color from the main key.
Basic operations in calculator
use AC button to clear all values.
To clear memory press shift then mode . Three screens will display as follow:
The magnitude of the induced EMF is not constant but depends on several factors, including the strength of the magnetic field, the speed of relative motion between the conductor and the magnetic field, the number of turns in the coil, and the area of the conductor exposed to the magnetic field. Understanding these factors is essential for improving the efficiency and performance of electromagnetic devices. This article examines the key factors that affect the magnitude of induced electromotive force and explains how each factor influences the amount of EMF generated.
Electromagnetic induction is the process by which an electromotive force (EMF) is generated in a conductor when it experiences a change in magnetic flux. This phenomenon, discovered by Michael Faraday, forms the basis of many electrical devices such as generators, transformers, and electric motors.
The amount of current produced from changing magnetic flux depends on a number of factors which includes:
Rate of change of magnetic flux
strength of magnetic field
number of turns in a coil
i. Rate of change of magnetic flux
The faster the rate of change of magnetic field, the higher the magnitude of the induced current.
Consider a coil of about 200 turns of a wire, sensitive galvanometer and a magnet arranged as shown in figure below.
To investigate how rate of change of magnetic flux, you move the magnet towards the coil and away at various speeds such as very fast, moderately fast and slowly.
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You observe that the faster the magnet is moved to and from the coil, the higher the deflection on the galvanometer. This shows that induced EMF is highest when the rate of change of magnetic flux is highest.
Magnetic flux could be interpreted as the number of magnetic field touching the coil at any given moment.
Explanations
Magnetic flux Φ is the strength of magnetic field threading a given area.
The magnetic flux Φ changes when the magnet is withdrawn from the coil where a faster withdrawal gives rise to a higher rate of change in magnetic flux linking the coil which then gives an increased induced Electromotive force(e.m.f)
see the diagram below that shows magnetic field lines:
ii. strength of magnetic field
Moving a stronger magnetic towards or away from the coil causes increase of the induced current when the speed of movement remains constant.
Consider a u-shaped electromagnet and a variable resistor connected to a circuit shown such that an electromagnet can have it’s strength varied by changing current passing through using the variable resistor.
After the setup, you can do the following to investigate the current induced with strength of the magnet:
Adjust the variable resistor so that minimum current flows.
Move the conductor PQ in a direction perpendicular to the magnetic field of the electromagnet and note deflection on the galvanometer.
change values of current and record corresponding readings on the galvanometer when wire cuts across the magnetic field.
Observations
Whenever current through the ammeter is increased, a greater deflection is obtained on the galvanometer when the conductor wire cuts across the magnetic field.
Explanations
Higher current passing through a coil of wire leads to a stronger electromagnet that will produce stronger magnetic field .
We can therefore conclude that the magnitude of the induced current is directly proportional to the strength of the magnetic field from which it is being produced.
iii. number of turns in a coil
If all other factors are held constant but the number of turns of wire on the coil increased, the induced current is observed to increase proportionately to increased number of turns.
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Having at your disposal insulated copper wire, sensitive galvanometer, magnet and connecting cables, you make a coil of numbered turns of wire and set up the apparatus as shown
to investigate how number of turns in a coil affects magnitude of the induced emf, do the following:
Insert a magnet in the coil and then withdraw it at a steady speed and then observe and record the maximum reading on the galvanometer.
Increase number of turns on the coil at equal intervals says 50, 100,150,200,250 etc and repeat the above procedure noting the maximum deflection each time.
Observations
Each time the number of turns of the coil is increased and all other factors held constant, a higher deflection on the galvanometer is recorded. The deflection is proportional to the number of turns used.
Explanations
Increased deflection indicates more current is produced in the coil. The induced emf is proportional to the number of turns and so we can say that each turn on the coil induces it’s own e.m.f. The total induced e.m.f is therefore a summation of all emfs produced by individual turns.
Infact by application of calculus, we can be able to express summation mathematically, but we will do that later in more advanced lessons.
Conclusions
Experiments shows that an e.m.f is induced in a circuit whenever magnetic flux linkage changes and the magnitude of the induced e.m.f increases with increase in the rate of change of the flux linkage and the number of turns of the coil.
The observations from experiments can be summarized in a Faraday’s law of electromagnetic induction which states that:
The magnitude of the induced e.m.f is directly proportional to the rate of change of magnetic flux linkage.
Revision Exercise
Electromagnetic Induction Quiz
Instructions: Answer all questions and click “Submit Quiz”.
The direction of the induced current is determined by Lenz’s Law. When a conductor or coil experiences a change in magnetic flux, an electromotive force (emf) is induced in it. This phenomenon is explained by Faraday’s Law of Electromagnetic Induction. However, Faraday’s Law does not indicate the direction of the induced current.
Lenz’s Law states that the induced current always flows in such a direction that the magnetic field produced by it opposes the change causing it.
In this experiment, a centre-zero galvanometer is first used to establish the relationship between the direction of current flow and the direction of galvanometer deflection. This relationship is then applied when studying induced currents in coils and magnets.
Experiment To Determine the Direction of the Induced Current in a Coil
Apparatus
Sensitive centre-zero galvanometer
Variable resistor (rheostat)
Dry cell
Switch
Connecting wires
Aim
To establish the direction of galvanometer deflection with respect to the direction of current flow.
Theory
Before studying induced currents, it is necessary to know which direction of galvanometer deflection corresponds to a particular direction of current flow.
The circuit shown consists of a battery, a rheostat, a switch and a sensitive centre-zero galvanometer connected in series.
When current flows from A to B, the galvanometer needle deflects in a particular direction. If the current is reversed and flows from B to A, the galvanometer deflects in the opposite direction.
This calibration enables the galvanometer to be used later in electromagnetic induction experiments.
Procedure
Set up the circuit as shown in the figure.
Adjust the variable resistor to a high resistance value.
Ensure that the switch is open.
Close the switch briefly.
Observe the direction of galvanometer deflection when current flows from A to B.
Reverse the battery terminals.
Close the switch again.
Observe the new direction of galvanometer deflection.
Record your observations.
Observations
Direction of Current
Galvanometer Deflection
A → B
Needle deflects to the right
B → A
Needle deflects to the left
Conclusion
The direction of galvanometer deflection depends on the direction of current flow.
This information can now be used to determine the direction of induced current when a magnet moves towards or away from a coil.
Virtual Laboratory Simulation
Switch Open
Investigating the Direction of Induced Current in a Coil
When a magnet moves relative to a coil, an electric current is induced in the coil. The direction of this current depends on whether the magnet is approaching or moving away from the coil. This simple experiment helps us understand the relationship between magnetic motion and induced current.
Procedure
Connect a sensitive galvanometer to a coil as shown in the experimental setup.
Move the north pole of a bar magnet toward the coil and observe the direction of the galvanometer’s deflection.
Next, move the magnet away from the coil and again note the direction of the pointer’s movement.
Observation
When the north pole of the magnet is moved toward the coil, the galvanometer pointer deflects to the left. This indicates that an induced current is flowing through the circuit in the direction D → C → B → A. see the figure below
When the north pole is moved away from the coil, the pointer deflects to the right. This shows that the induced current now flows in the opposite direction, D → A → B → C.
Explanation
As the north pole of the magnet approaches the coil, the magnetic flux linked with the coil increases. An induced current is therefore produced in the coil. The direction of this current is such that the coil behaves like an electromagnet with its north pole formed at the end nearest to the incoming magnet.
The induced north pole repels the approaching north pole of the magnet. In this way, the induced magnetic field opposes the change that produces it.
When the magnet is moved away from the coil, the magnetic flux through the coil decreases. The induced current reverses its direction, causing the end of the coil nearest the magnet to become a south pole. This south pole attracts the receding north pole of the magnet.
Again, the induced magnetic field acts to oppose the change in magnetic flux. This behavior is consistent with Lenz’s Law, which states that the induced current always flows in a direction that opposes the cause producing it.
Direction of Induced Current in a Straight Conductor
Lenz’s law can also be used to determine the direction of induced current in a straight conductor moving through a magnetic field. Consider a conductor AB placed between the poles of a strong U-shaped magnet as shown in below.
When the conductor is moved across the magnetic field, it cuts the magnetic lines of force and an e.m.f. is induced in it.
As the conductor is moved upwards, the magnetic flux linked with the conductor changes. According to Lenz’s law, the induced current flows in such a direction that the magnetic effect produced opposes the upward motion. It is observed that the current flows from B to A as shown. The
The induced current flows into the page, away from the observer’s eye(B to A)
The induced magnetic field therefore acts to oppose the motion responsible for its production.
When the conductor is moved downwards, the change in magnetic flux occurs in the opposite direction. Consequently, the induced current reverses and flows from A to B as shown.
Induced current out of page (flowing A to B)
The magnetic field produced by this current opposes the downward motion of the conductor.
This experiment demonstrates an important feature of electromagnetic induction: whenever the direction of motion is reversed, the direction of the induced current also reverses. The induced current is always such that its magnetic effect opposes the change that produces it.
The opposition predicted by Lenz’s law is a direct consequence of the principle of conservation of energy. Mechanical work must be done to move the conductor through the magnetic field. This mechanical energy is converted into electrical energy in the conductor. If the induced current aided the motion instead of opposing it, energy would be produced without any external work being done, which would violate the law of conservation of energy.
The direction of the induced current may also be determined using Fleming’s Right-Hand Rule. The forefinger is pointed in the direction of the magnetic field, the thumb in the direction of motion of the conductor, and the middle finger then indicates the direction of the induced current. The direction obtained using Fleming’s rule is always consistent with Lenz’s law.
Interactive Simulation: Lenz’s Law in a Moving Conductor
The following simulation is designed for direct embedding in a WordPress Custom HTML block. It demonstrates a conductor moving vertically through a magnetic field between the poles of a U-shaped magnet. The induced current direction changes automatically according to Lenz’s law.
Electromagnetic Induction: Moving Conductor AB
Motion:Stationary
Current:None
Galvanometer:0
What Is Fleming's Right-Hand Rule?
Fleming's Right-Hand Rule is used to determine the direction of induced current in a conductor moving within a magnetic field. It is a simple yet powerful tool used to determine the direction of induced current in a conductor moving through a magnetic field. Based on the principles of electromagnetic induction, the rule has practical applications in generators, renewable energy systems, and science education. By relating the directions of motion, magnetic field, and current, it helps explain one of the most important processes in modern electrical technology.
Fleming's Right-hand rule states that:
If the thumb and the first two fingers of the right hand are held mutually at right angles with the first finger pointing in the direction of the field, thumb pointing in the direction of motion, then the second finger points in the direction of the induced current.
The rule is mainly applied in electric generators, where mechanical energy is converted into electrical energy.
To use the rule, the thumb, forefinger, and middle finger of the right hand are held at right angles to one another.
Each finger represents a different quantity:
Thumb – Direction of motion of the conductor.
Forefinger – Direction of the magnetic field (from north to south).
Middle finger – Direction of the induced current.
When these three fingers are positioned correctly, the middle finger points in the direction of the current generated in the conductor. From the diagram above part b, Fleming's right rule predicts that the induced currents flows from X to Y. The current must flow in the direction X-Y in order to produce a force to the left opposing the motion to the right.
Example 1
A square loop of a conductor is pulled at a steady speed across a uniform magnetic field as in figure below.
(a) Determine in the figure the direction of induced current in the sides AB, AD, CD and BC, if any.
(b) Explain what happens when:
(i) all the sides are moving in the uniform field and state the potential difference across points AB.
(ii) the side CD leaves the field.
(c) Suggest why in the absence of friction, more force is required to keep the coil moving at a steady speed when side CD leaves the field.
solution
(a)
Sides AD and BC have no induced e.m.f. and hence no induced current, since they are not cutting the magnetic field. Sides AB and CD cut the magnetic field, causing current to flow from B to A in AB and C to D in CD.
(b)
(i) The currents in AB and CD are equal in magnitude and opposite each other. The resultant potential difference across the points A and B is zero.
(ii) Induced e.m.f. in AB sets up a current that takes the path A → D → C → B → A. There is no induced current in side CD.
(c)
The flow of current in AB creates a force that tends to oppose the motion.
Fleming's right hand rule which is also known as dynamo rule is in agreement with the Lenz's law.
The Science Behind the Rule
Fleming's Right-Hand Rule is based on the principle of electromagnetic induction discovered by Michael Faraday in 1831. Faraday found that when a conductor moves through a magnetic field, an electromotive force (EMF) is induced in it. This phenomenon is known as electromagnetic induction.
The induced current flows because the moving conductor cuts across magnetic field lines. The direction of this current depends on both the direction of motion and the direction of the magnetic field. Fleming's Right-Hand Rule provides a simple method for determining this direction without complex calculations.
Phase difference is the angular difference between two sinusoidal waveform of the same frequency and It tells us how much one wave is “ahead of” or “behind” another in terms of their cycles
The word phase in normal usage means any stage in a series of events or in a process of development.
Cambridge University dictionary defines phase as one of the stages or points in a repeating process measured from a specific starting point.
Two Waves can be of the same amplitude but with the different frequencies as shown in figure below.
waves of the same amplitude but different frquency
The wave profile P makes it’s one complete oscillations before wave Q. Wave P has shorter wavelength compared to Q and hence P has higher frequency.
We can also see that P has smaller period as waves with shorter wavelength has smaller period.
Wave P completes it’s first cycle at A while Q finishes it’s first oscillation at B. We can say that P is leading Q. The maximum displacement of the two waves is the same hence they are operating at the same amplitude but different frequency. The two waves are said to be out of phase. Think about two radio receivers tuned to two different stations but with equal volume.
waves can also be of the same frequency but different amplitudes. Think of when we tune in our two radio receivers to the same station and then set them at different volumes
The figure below illustrates two waves operating at same frequency but at different amplitudes.
Waves of the same frequency but different amplitude
One wave is having more displacement than the other. However, they are arriving at the horizontal position simultaneously, as can be seen from the diagram. We say they are in phase.
Pendulum bobs in phase
To further illustrate the concept of phase and out of phase oscillations, consider two identical pendulums. The pendulums have bobs P and Q below.
Masses oscillating in phaseIn-Phase Pendulums
We set the two masses, P and Q in oscillation. We give them some displacement on the left and then releasing them simultaneously. They have equal displacement because we have released at the same time. Therefore, they will pass through the lowest point Y simultaneously as they move in the opposite direction. They attain displacement together at Z and swing back together to complete the oscillation at x.
At any particular moment, the two masses will be moving in the same direction. They will also be at the same level of displacement in their oscillation. We say that the masses are oscillating in phase.
particles in phase difference
When particles in a wave motion happens to be oscillating in the same direction and at the same level of displacement,we say that their oscillation are in phase.
The diagram below have highlighted two positions of particles A and B. The particles are at the same displacement level from the reference line. They are both facing the same direction as indicated by the arrows. The particles A and B are said to be in phase and their distance apart is the wavelength λ of the wave motion whereas time taken to move from A to B is the periodic time T.
Particles in a wave motion can be in phase even if they have different amplitude.
In our previous pendulum oscillation of mass P and Q ; If P is Initially given a larger displacement than Q, the two will oscillate in phase. However, P will always be at a larger magnitude of displacement than Q.
A typical displacement time graph for two wave motions in phase with different amplitudes is shown below.
two waves in phase at different amplitude.
Oscillations out of phase
Consider two masses P and Q displaced from opposite directions from each other as in figure below.
When released simultaneously, they pass through the rest position at the same time. They move in opposite directions. They reach a point of maximum displacement at the same time. However, their maximum displacement is in opposite directions to each other.
Waves 180o out of phase difference
The two objects above are always at the opposite levels of displacement and their oscillations opposite direction to each other and they are said to be in opposite phase.
Wave motions that have same displacement and makes complete oscillations at the same time with their maximum displacements in exact opposite to each other are said to be in 180o phase difference (180o out of phase).
The figure below shows two wave motions at 180o phase difference.
Waves 90o out of phase difference
suppose in our pendulum oscillations we displaces the objects P and Q to X ; we release Q before P and then we release p when Q is exactly at Y. The angle of oscillation between P and Q will be 90o in difference and the resulting oscillation will be 90o out of phase.
The displacement time graph for waves 90o out of phase is illustrated below.
two waves can be out of phase at any angle. We are to see that in our future lessons.
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