Electric current and potential difference represents two phenomenon that depends on each other to exist in electricity concepts. An electric current is the rate of flow of charge through a conductor. Current flows when there is a potential difference between two points in a conductor. Electric current is measured in amperes by an instrument called ammeter. An ammeter is an electrical instrument used to measure the current flowing through a circuit. The ammeter is designed to be connected in series with the circuit. This ensures that the current flows through the ammeter, allowing it to accurately measure the amount of electrical current.

There are two types of ammeters:

Instruments used in experiments of electric current and potential difference

• Analog Ammeter: This uses a needle or pointer to indicate the current on a scale.

The figure below shows an analog ammeter ammeter common in school laboratories.

2. Digital Ammeter: This displays the current measurement on a digital screen. It provides a digital readout of the electrical current. Digital ammeter allows one to choose a scale of measurement in amperes (A), milli-amperes (mA), or micro-amperes (µA). It uses a numerical display rather than a moving needle or pointer.

Electric current flows between two points in a closed path due to a potential difference between those two points. Sometimes the flowing current can be too small to me measured by an ammeter. A more sensitive instrument may therefore be required to measure small currents.

A millimeter is an instrument used to measure current in terms of one in thousand of an ampere. A milliammeter measures current in terms of milli-amperes.

Much smaller currents can be measured by a micro- ammeter. A micro-ammeter measures current in terms of micro-ampere.

Using an ammeter in measuring electric current

An ammeter has very low electrical resistance. Therefore it is connected in series with the instrument whose current passing through need to be measured. When connecting an ammeter in the circuit, ensure it is done correctly. The correct procedure is such that current enters the ammeter through positive terminal and exits through the negative terminal. If connected such that convectional current enters through negative terminal, the ammeter may get damaged.

The figure below shows the ammeter connected in series with the bulb. The convectional current flowing through the bulb also flows through the ammeter.

The figure below shows wrong ammeter connection. Note that the positive terminal of the ammeter is connected to the negative terminal of the cell.

Before connecting the ammeter in a circuit, confirm that it’s pointer is at zero mark on the scale. Otherwise, use the zero adjusting screw to move it to the correct position. Most of ammeters has two scales. An appropriate scale should be selected to safeguard the coil from damaged if current passing exceeds its capacity. For example an ammeter can have a scale of (0-3)A or (0-5)A. The figure below shows an ammeter dashboard with two scales; (0-5)A and (0-2.5)A.

If a scale of (0 – 5) A is selected, the meter can read up to 5 A. With such a scale, 10 divisions represents 1.0 A. For a (0-2.5) A scale, ten divisions will represent 0.5 A meaning each division is 0.05 A. From the diagram, the reading on the ammeter is 2.45 A while reading (0-5) A or 1.225 while reading the (0 -2.5) A.

Electric current and potential difference: using a voltmeter

while investigating electric current and potential difference, we need to measure potential difference across various components in the circuit. A voltmeter is always connected across the device (parallel to the device) which the voltage is to be measured. The figure below shows voltmeter connected across the bulb in parallel arrangement.

Voltmeter is connected in series because it is an instrument with high resistance to the flow of current. Therefore, It takes no current from the component across which the voltage is to be measured.

The positive terminal of the voltmeter is connected to the point where convectional current is entering a component. Its negative terminal is connected to the point where the current is leaving the component.

One should ensure that the pointer is exactly on the zero mark before connecting the voltmeter. If pointer is not at zero, the pointer should be adjusted to zero by the screw.

potential difference

Work must be done to move an electric charge through a conductor. The device that produces energy to do this work is called a source of electromotive force (e.m.f). The source may be a battery, which converts chemical energy to electrical energy, or a generator, which converts mechanical energy to electrical energy. When the battery does the work of pumping charges through a conductor or an electrical device, an electrical potential difference(p.d) develops between its end. This potential difference is measured in volts using the voltmeter.

Potential Difference and the Voltmeter

A lack of “pumping” charges through a conductor indicates that there is no potential difference between two points. The potential difference (p.d) between two points A and B (V_ab)of a conductor is defined as the work done in moving a unit charge from point B to A.

in other words:

where: (V_AB) = potential difference across AB.

• (W) = work done (in joules)
• (Q) = charge moved (in coulombs)

From the equation, one volt is equal to one joule per coulomb.

Measuring potential difference

The voltmeter measures potential difference. In laboratories, moving coil voltmeters are commonly used, although today many of these instruments are replaced by digital voltmeters.

(a) Analogue voltmeter

(b) Digital voltmeter

Please note that instruments like fuel gauge and speedometers are essentially voltmeters.

Example

In moving a charge of 10 coulombs from point B to point A, 120 joules of work is done. What is the potential difference between A and B?

solution:

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Using a Voltmeter

A voltmeter is always connected across (in parallel to) the device whose voltage is to be measured. consider the diagram below.

Because the voltmeter has a high resistance to the flow of current, it draws very little current from the component.

The positive terminal of the voltmeter is connected to the point where conventional current enters the component, while the negative terminal is connected to the point where the current leaves the component.

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Experiment To Investigate the Current and Voltage in a Parallel Circuit Arrangement

Apparatus

• Two 1.5 V cells
• 3 identical bulbs
• 3 ammeters
• 4 voltmeters
• Switch
• Connecting wires

Procedure

• Connect the circuit as shown in figure below.

• Switch on the circuit and take the readings on the ammeters A₁, A₂,A₃ and A₄.
• Switch off the circuit and disconnect the ammeters.
• Connect the bulbs and the voltmeters as shown in figure

• Take the readings on V₁,V₂,V₃ and V₄.

Observation

1. Reading on A₁ = Reading on A₂ + Reading on A₃ = Reading on A₄
2. Reading on V₁ = Reading on V₂ = Reading on V₃ = Reading on V₄.

Conclusion

When components are connected in parallel:

1. The sum of the currents in parallel circuits is equal to the total current. The total current entering a junction equals the total current flowing out.
2. The same voltage drops across each of the components.

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Example 2

Find the current passing through L₁ in figure below, given that 0.8 A passes through the battery, 0.28 A through L₂, and 0.15 A through L₃.

Solution

Current through battery = Current through L1 + Current through L2 + Current through L3

0.8 = I₁+I₂+I₃

0.8=I₁ + 0.28+0.15

0.8 =I₁ + 0.43

Therefore:

I₁ = 0.8A – 0.43A = 0.37A

Experiment To Investigate Current and Voltage in Series Arrangement

Apparatus

• 4 voltmeters
• 3 torch bulbs (2.8 V)
• Bulb holder
• Switch
• Connecting wires
• Two cells

Procedure

• Connect the circuit as shown in figure below

• Switch on the circuit and record the readings on the meters.
• Switch off the circuit and disconnect the ammeter.
• Connect the bulbs, ammeter, and voltmeters as shown in figure below.

• Switch on the circuit and record the readings on the meters.

observations

The reading of current by the ammeters A₁ and A₂ and A₃ is the same.

The total voltage drops across the bulbs (V₁+v₂+v₃) equals to the total voltage v₄ across the terminals of the battery.

please note that the observations will remain true even when the bulbs are not identical.

conclusions

In a series arrangement, the same current flows through each component.

The sum of the voltage drops across the components is equal to supply voltage

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Summary

• A voltmeter measures potential difference.
• Potential difference is the work done per unit charge.
• Voltmeters are connected in parallel across components.
• In parallel circuits, current splits while voltage remains the same.
• In series circuits, current remains the same throughout the circuit.

Related Topics

• Source of Electric current: precise study
• Electromagnetic induction
• Arrangement of cells: Easy introduction
• Electromotive force and potential difference: introduction
• Installing Git on windows
• Trigonometry

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Examination questions on measurements basically covers concepts like:

•  Length and Distance
• Weight and Mass
• Basic Units and Conversions
•  Volume and Capacity
• Time
• Temperature
• Derived Units and Calculations
•  Measuring Instruments
• Precision and Accuracy

Here are the questions that involves measurements

1. The diagram below shows a piece of wood whose length is being measured using a strip of           measuring tape.

            What is the length of the piece of wood.

2. Figure 1 below shows a Vernier calipers being used to measure the thickness of an object. It has a error of +0.01 cm.

What is the correct measurement?                                                                       (2 marks)

3. Figure  below shows  Perspex  container  with a square base  of side  5 cm . It is carrying  water  to a height  of 7 cm.

When pebble is immersed  into  the water, the level  rise  to 10 cm. what is  the volume of the pebble? (2 marks)

4.   A  drop  of  oil volume  6 x 10 ^-9 m³ forms  a patch  of area  0.0755 m²  on a water  surface. Estimate the of an oil  molecule ( 2 marks).

Related topics

• Introduction to measurement error
• Basic Physical quantities

The table of tangents holds values for every acute angle from 0^o to 90^o. 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 0^o to 90^o. 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 (1^o) 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 0^o to 90^o  and has column head labeled x^o which represents

The second column consists of 0.0^o to 0.9^o which divides a degree into 10 smaller units hence giving an accuracy of 0.1^o.

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 36.57^o

solution

In the column labelled x^o , 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 77^o48′

solution

1^o=60′, hence 48′ = (48′ x 1^o)/60′ = 0.8^o

then 77^o48′ can be expressed as 77.8^o

From the tables, you identify row 77 at x^o 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 77^o48′ = tan 77.8^o = 0.6252

Example

Find angle θ and α in the figure below.

Related Topics

• Table of tangents
• Equations of a circle
• Trigonometry
• Tangent of an angle
• Electromagnetic induction

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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:

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) = x² and

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;

and hence

and finally

Revision Exercise

Answers to revision exercise

Related topics

• Basic Integration rules
• The second fundamental theorem of calculus

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