1. The document discusses electric charge and electric fields. It defines electric charge, quantization of charge, and conservation of charge.
2. Coulomb's law states that the electrical force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
3. An electric dipole is a system of two equal and opposite charges separated by a distance. The electric field of a dipole depends on the relative positions of the two charges.
Electrostatic potential and capacitanceEdigniteNGO
Hello everyone, we are from Edignite NGO and we have come up with chapters of class 11 and 12 (CBSE).
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Electrostatic potential and capacitanceEdigniteNGO
Hello everyone, we are from Edignite NGO and we have come up with chapters of class 11 and 12 (CBSE).
For any queries, please contact
Lekha Periwal : +916290889619
Heer Mehta : +917984844099
These slides describe Gauss's Law in electrostatics and its application to find out electric field due to point charge, uniformly charged spherical shell, and electric potential due to spherical shell.
Visit this link: https://phystudypoint.blogspot.com
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These slides describe Gauss's Law in electrostatics and its application to find out electric field due to point charge, uniformly charged spherical shell, and electric potential due to spherical shell.
Visit this link: https://phystudypoint.blogspot.com
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
VISIT US @
www.anuragtyagiclasses.com
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
VISIT US @
www.anuragtyagiclasses.com
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
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CURRENT ELECTRICITY/ELECTROSTATICS FOR CBSE FREE REVISION SHEET BY ANURAG TY...ANURAG TYAGI CLASSES (ATC)
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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Electric Charge and Electric Field LectureFroyd Wess
More: http://www.pinoybix.org
Lesson Objectives:
Static Electricity; Electric Charge and Its Conservation
Electric Charge in the Atom
Insulators and Conductors
Induced Charge; the Electroscope
Coulomb’s Law
Solving Problems Involving Coulomb’s Law and Vectors
The Electric Field
Field Lines
Electric Fields and Conductors
Gauss’s Law
Electric Forces in Molecular Biology: DNA Structure and Replication
Photocopy Machines and Computer Printers Use Electrostatics
This is first PPT in the electrostatics series. This PPT presents idea of charge , its various methods of production like through conduction, friction, induction. It also describes working of electroscope & concept of grounding of an insulator.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
*Animated PPT FOR SCHOOL/ Coachings *
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JEE Main Advanced 12 Sample ebook, which helps you to understand the chapter in easy way also download sample papers and previous year papers and practice to solve the question on time. Download at www.misostudy.com.
JEE Main 12 Sample ebook, which helps you to understand the chapter in easy way also downaload sample papers and previous year papers and practice to solve the question on time. Download at www.misostudy.com.
This document has the physics most important topics type wisew and chapterwise .
the document has been created by physics guru ANURAG SIR .
AFTER DOING THESE TOPICS , A STUDENT CAN SCORE MORE TAHN 90 % ONLY IN ONE MONTH IN PHYSICS BOARD EXAM .
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!SOFTTECHHUB
As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
The Art of the Pitch: WordPress Relationships and Sales
01(T) Electric Charge And Electric Field
1. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 1
1.1 Electric Charge
• Of almost more than 100 fundamental particles of matter, three most important are electron,
-31 -27
proton and neutron. Their masses are m e = 9.1 × 10 kg, m p ≅ m n = 1.6 × 10 kg
respectively.
-67
• Gravitational force of attraction between two electrons 1 cm apart is 5.5 × 10 N, whereas
-24
electrical force of repulsion due to electric charge on them is 2.3 × 10 N which is much
stronger.
• Electric charge can be positive or negative. Traditionally, charge of proton is considered
positive and that of electron negative although reverse sign convention would have made
no difference.
• Like charges repel each other and unlike charges attract. Electroscope is used to detect
charges.
• Electrons revolving around the nucleus are weakly bound as compared to the force with
which protons are bound inside the nucleus. Hence, during exchange of electrons between
two bodies, electrons get transferred from one body to the other.
• SΙ unit of charge is coulomb denoted by C. It is the charge passing in 1 second through
Ι
any cross-section of a conductor carrying 1 ampere current. Magnitude of charge on an
-19
electron or a proton is 1.6 × 10 C.
• Electric charge, like mass, is a fundamental property which is difficult to define.
1.2 Quantization of Electric Charge
The magnitudes of all charges found in nature are in integral multiple of a fundamental
charge ( Q = ne ). This fact is known as the quantization of charges. This fundamental charge
is the charge of an electron and is denoted by e.
All fundamental charged particles possess charge having magnitude e. For example, a proton
and a positron ( positive electron ) possess positive charge ( +e ). Atom as a whole is
electrically neutral as there are equal number of protons and electrons in it. This fact has
20
been verified with an accuracy of 1 in 10 .
No theory has been able to satisfactorily explain the quantization of charges so far.
Protons and neutrons are believed to be made up of more fundamental particles called
/
quarks. Quarks are of two types; ‘up quark’ possessing +( 2/3 )e charge and ‘down quark’
/
possessing -( 1/3 )e charge. The independent existence of quarks is not detected so far.
1.3 Conservation of Electric Charge
Irrespective of any process taking place, the algebraic sum of electric charges in an
electrically isolated system always remains constant. This statement is called the law of
conservation of charge.
In an electrically isolated system, a charge can neither enter nor leave it. Any charge-less
matter or radiation can enter or leave the system. γ - ray photon entering the system may
2. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 2
produce an electron-positron pair which as a whole being electrically neutral does not alter
the original charge of the system.
1.4 Charging by Induction
If two identical spheres, one carrying electric charge Q and the other no charge, are brought
in contact and separated, each will possess equal charge Q / 2 after separation. Thus an
uncharged sphere gets charged. Another method of charging a substance is explained as
under.
Fig. 1 shows a sphere with zero charge.
Fig. 2 shows a plastic rod rubbed with fur which acquires negative charge brought close to
the sphere. This repels free electrons on the sphere to a part away from the rod
leaving part of the sphere closer to the rod positively charged.
Fig. 3 shows the electrons on the sphere conducted to the earth by earthing the sphere.
Fig. 4 shows that the positive charge is still retained by the sphere even on removal of the
earthing.
Fig. 5 shows electrons redistributed on the sphere so that the positive charge is spread all
over the surface of the sphere.
This shows that a body can be charged without bringing in physical contact with another
charged substance. This phenomenon is called induction of electric charge.
1.5 Coulomb’s Law
“The electrical force ( Coulombian force, F ) between two stationary point charges ( q1 and q2 )
is directly proportional to the product of the charges ( q1q2 ) and inversely proportional to the
2
square of the distance ( r ) between them.” This statement is known as Coulomb’s law.
q q q q 1 q q
F ∝ 1 2
⇒ F = k
1 2
⇒ F =
4 πε
1 2
r2 r2 0 r2
9 2 -2
where F is in N, q1 and q2 are in C, r is in m and k = 9 × 10 Nm C in vacuum is the
- 12 -1 -2
proportionality constant. ε0 = 8.9 × 10 2
C N m is the electrical permittivity in vacuum.
If the charges are in medium other than vacuum, then the electrical permittivity of the
medium, ε, should be used in the above equation in place of ε0 . The ratio ε / ε0 is called
relative permittivity, εr, of that medium. The Coulomb’s law for any medium is written as
1 q q
1 2
F = , where ε = ε0 εr
4 πε r 2
3. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 3
Coulomb’s Law in Vector Form
Let qi and qj be two electrical like charges ( both positive or
→ →
both negative ) having position vectors ri and r j respectively
→
in a Cartesian co-ordinate system. The force, Fij , acting on
charge qi due to qj, directed from qj to qi, is given by
→ →
→ qi q j ri - rj qi q j → →
F ij = k = k ( ri - rj )
→ → → → → →
l ri - r l2 l ri - r l l ri - r l3
j j j
→ →
ri - rj
where is the unit vector in the direction from qj to qi .
→ →
l ri - r l
j
→
Similarly, the force, Fji , acting on charge qj due to qi, directed from qi to qj, is given by
→ →
→ qi q j rj - r
i
qi q j → →
Fji = k
→ → → → = k
→ → 3
( rj - r )
i
2
lr - r l l rj - r l l rj - r l
j i i i
→ →
rj - ri → →
where is the unit vector in the direction from qi to qj . Note that F ij = - F ji .
→ →
l rj - r l
i
1.6 Forces between More than Two Charges: The Superposition principle
“ When more than one Coulombian force are acting on a charge, the resultant
Coulombian force acting on it is equal to the vector sum of the individual forces.”
Consider charges q1, q2 and q3 having position vectors
→ → → → →
r1 , r2 and r3 respectively. Let F21 and F23 be the forces
acting on charge q2 due to charges q1 and q3 respectively.
→ q 2 q1 → →
Then, F21 = k
→ → 3
( r2 - r )
1
and
l r2 - r l
1
→ q q3
2
→ →
F23 = k
→ → 3
( r2 - r )
3
l r2 - r l
3
and from the principle of superposition, the resultant force acting on charge q2 is
4. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 4
→ q 2 q1 → → q2 q3 → →
F2 = k
→ → 3
( r2 - r )
1
+ k
→ → 3
( r2 - r )
3
and in short,
l r2 - r l l r2 - r l
1 3
→ 3 qj → →
= ∑ ( r2 - r )
→ →
F2 kq2
j=1 j
j≠2 l r2 - r j l3
In general, the force acting on charge qi due to a system of N electric charges will be
→ N qj → →
Fi = kq ∑ ( ri - r )
i j = 1 → → j
j ≠ i l ri - r
j
l3
1.7 Continuous Distribution of Charges
The continuous distribution of charges can be of three types:
( 1 ) Line Distribution, ( 2 ) Surface Distribution and ( 3 ) Volume Distribution
Line Distribution
→
Let r' = position vector of a point on the curved line
as shown in the figure,
→
λ ( r ' ) = linear charge density at the above point,
→
dl' = length of a small line element at that point,
then the amount of charge in that line element
→ →
= λ ( r ' ) l dl' l and
the electrical force acting on charge q having position
→ →
→ → kq λ ( r' ) l dl' l → →
vector r is given by dF = ( r - r' )
→ →
lr - r' l 3
→ →
→ → →
λ ( r' ) l dl' l
On integrating, total force F = kq
∫lr
l
→ →
- r' l3
( r - r' )
Surface Distribution
→
Let σ ( r ' ) = surface charge density at a point having position
→
vector, r ' , on any surface,
→
da' = area vector of a small area around that point as
shown in the figure,
5. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 5
→
On calculating the force acting on any charge q having position vector, r , due to the
→
charge in the small surface element, da' , and integrating over the entire surface we get total
force
→ →
→ σ ( r' ) l da' l → →
F = kq
∫ →
lr -
→
r' l 3
( r - r' )
a
Volume Distribution
→
Let ρ ( r' ) = volume charge density at a point having
→
position vector, r ' , in any volume,
→
dV ' = small volume element of the entire volume,
On calculating the force acting on any charge q having
→
position vector, r , due to the charge in the small
→
volume element, dV ' , and integrating over the entire
volume we get total force
→ →
→ ρ ( r' ) l dV' l → →
F = kq
∫ lr → →
- r' l3
( r - r' )
V
1.8 Electric Field
The region around a system of charges in which the effect of electric charge is prevailing is
called the electric field of that particular system of electric charges.
“ The force acting on a unit positive charge at a given point in an electric field of a point
charge or of a system of charges is called the electric field ( or the intensity of electric
→
field ) E at that point.”
→ →
→ → F( r) N qj → →
Thus, E( r) = = k ∑ ( r - rj )
q j=1 → →
l r - r j l3
Here, q1, q2, ....., qN are the sources of the electric field.
-1 -1
The unit of electric field intensity in SI system is N C ( or V m ).
Noteworthy points for an electric field
1 ) The electric charge used to measure electric field intensity is called a test charge.
2 ) If we know electric field intensity at all the points in the electric field, there is no need to
know the source charges or their locations in the field.
6. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 6
3) The test charge should be as small as possible to ensure that its presence makes no
change in the original field.
4 ) The direction of force experienced by a positive charge at any point is the direction of
electric field at that point.
5 ) Faraday first introduced the concept of an electric field which is a physical reality.
1.8 ( a ) Electric Field due to a Point Charge
Taking the position of charge Q as origin, the electric force due to it acting on charge q at
a distance r from it will be
→ kQq
F =
r2
Hence, electric field
intensity due to charge
Q will be,
Q<0 Q>0
→
→ F kQ
E = =
q r2
The figure shows the electric
field due to point charges in
two dimensions. Actual field
spreads radially in all
directions intersecting spherical surface perpendicularly, centre of the sphere located at the
point charge, and is directed outwards if the charge is positive and inwards if the charge
is negative.
The strength of the electric intensity decreases away from the charge as indicated by
decreasing length of arrows.
The electric field due to more than one charge is equal to the vector sum of the individual
electric fields due to all the charges.
1.9 Electric Dipole
A system of two equal and opposite charges, separated by a finite distance, is called an
electric dipole. If the charges are q and -q and 2a is the distance between them, electric
dipole moment of the dipole is
→ →
p = q(2a )
Electric dipole moment is a vector quantity and its direction is from the negative electric
charge to the positive electric charge. Its unit is coulomb-meter ( Cm ).
The total charge on an electric dipole is zero, but its electric field is not zero, since the
position of the two opposite charges is different.
1.9 ( a ) Electric Field of a Dipole
To find the electric field of a dipole, let origin of the co-ordinate system be at its mid-point.
Let the +q charge be on positive Z-axis and -q charge be on negative Z-axis and the
separation between them be 2a.
7. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 7
→ →
The position vector of +q charge is r1 ( 0, 0, a ) and that of -q charge is r2 ( 0, 0, -a ). The
→
electric field due to this dipole at any point having position vector r is given by
→ → → →
→ → ( + q ) ( r - r1 ) ( - q ) ( r - r2 )
E( r ) = k +
→ → → →
3 3
l r - r1 l
l r - r2 l
For a point z on z-axis
-
→
having position vector, r = ( 0, 0, z ),
→ q ( 0, 0, z - a ) q ( 0, 0, z + a )
E (z) = k -
l ( z - a ) l3 l ( z + a ) l3
q -q ^
= k + p
(z - a) 2 (z + a)
2
kq ( 4za ) ^
= p
2 2 2
(z - a )
But 2aq = p,
→ 2kpz ^ 2kp ^
∴ E (z) = p = p
2
( ignoring a compared to z
2
if z > > a )
(z
2
- a )
2 2
z3
→
For a point y on y-axis having position vector, r
- = ( 0, y, 0 ),
→ q ( 0, y, - a ) q ( 0, y, a )
E (y) = k -
3 3
( y2 + a2 ) 2
( y2 + a 2 ) 2
kq k q( 2a ) ^
= ( 0, 0, - 2a ) = - p
3 3
2 2 2 2 2 2
(y + a ) (y + a )
kp ^
= - p
3
2
(y + a2 ) 2
kp ^
= - p ( if y > > a )
y3
8. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 8
1.10 The Behaviour of an Electric Dipole in a Uniform Electric Field
→ →
An electric dipole p = q ( 2 a ) is kept in a uniform electric field
→
E making an angle θ with it. The origin of the co-ordinate
→
system, O, is at the centre of the dipole and E is directed along
the positive Y-axis.
The resultant of qE and -qE forces acting on +q and -q charges
respectively being zero, the dipole is in translational equilibrium.
But as the two forces have different lines of action, the dipole
will experience a torque.
The torques acting on charge +q due to force qE and on charge
-q due to force -qE respectively with respect to origin are
→ → → → → → → →
τ = a × q E and τ = (- a ) × (-q E ) = a × q E
1 2
The total torque acting on the dipole is
→ → → → → → → → →
τ = τ + τ = ( a × q E ) + ( a × q E ) = 2q a × E
1 2
→ → →
∴ τ = p × E ( in anti-clockwise direction )
The magnitude of this torque is τ = pE sin θ and its direction is perpendicularly coming out
of the plane of figure.
The dipole rotates due to this torque till the angle θ reduces to zero and the dipole aligns
itself along the direction of the electric field. This is the equilibrium position of the dipole
( about which dipole oscillates in absence of damping ) and if it has to be rotated by some
angle from this position, work will have to be done equal to the change in potential energy
of the dipole.
1.11 Behaviour of an Electric Dipole in a Non-uniform Electric Field
In a non-uniform electric field, the intensity of the
field being different at different points, different
forces act on the two charges of the dipole. Hence
the dipole experiences a linear displacement in
addition to rotation.
→
Let the electric field intensity be E at -q charge
and let it increase linearly in the X-direction. Let
the x-coordinate of -q charge be x. Then from the
figure, x-coordinate of +q charge is x + 2a cos θ.
Also the electric intensity near +q charge will be
dE
E + 2a cos θ. The electrical force acting on
dx
9. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 9
→
→ → dE
these charges will be -q E and + q(E + 2a cos θ ) .
dx
→
dE
The net force on the dipole being q 2a cos θ , the dipole will have acceleration in the
dx
positive x-direction in addition to rotation in the clockwise direction. The rotation will stop
when the dipole aligns in the direction of the field ( assuming damping is present ) but the
translation will continue in the positive x-direction.
When a dry comb charged by rubbing with dry hair is brought close to small pieces of
paper, electric dipole is induced in them in the direction of non-uniform electric field. This
exerts a net force on the pieces of paper which get attracted to the comb.
1.12 Electric Field Lines
Michael Faraday introduced the concept of electric field lines and called them lines of force.
An electric field line is a curve drawn in the electric field in such a way that the tangent to
the curve at any point is in the direction of electric intensity at that point.
1.12 ( a ) Characteristics of Electric Field Lines
(1) The tangent drawn at any point on the electric field line indicates the direction of
electric intensity at that point.
(2) Two electric field lines do not intersect because if they do then two tangents can be
drawn at their point of intersection which is not possible.
( 3 ) The distribution of electric field lines in the region of the electric field gives the
intensity of electric field in that region.
The number of electric field lines passing perpendicularly through unit cross-sectional
area about a point is proportional to the electric intensity at that point. Hence, the field
lines will be crowded where the electric intensity is more and sparse where it is less.
Let there be N ( an arbitrary number ) number of field lines
perpendicular to the surface of a sphere of radius R due to a
point charge q as shown in the figure. This is not the flux.
Now, the number of field lines per unit area is proportional to
the electric intensity.
N 1 q βq
∴ ∝ ∴ N =
ε0
4 πR
2 4πε 2
0 R
where, β is the proportionality constant value of which can be determined from the
initially assigned arbitrary number, N.
In the case of an electric dipole, the number of field lines originating from +q charge
enter into -q charge as both the charges are of the same magnitude. But if one charge
is q and the other is -q’, where q > q’, then the number of electric field lines leaving
the charge +q will be
10. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 10
βq
N =
ε0 and the number of electric field
lines entering -q’ charge will be
β q'
N' =
ε0 .
Thus out of N number of lines N’ number of
lines enter the charge -q’ and the remaining
lines become radial at large distances and
move to infinity as shown in the figure.
The electric field lines are used for
geometrical representation of electric field
and are imaginary. The electric field is a reality.
(4) The field lines of a uniform electric field are mutually parallel and
equidistant.
(5) The field lines of a stationary electric charge do not form close loops.
The adjoining figure shows electric field lines of an electric dipole.
1.13 Electric Flux
Consider an arbitrary surface in an electric field as shown in the figure. An infinitely small
element of it can be considered flat if its surface is not highly irregular. It can be considered
as a vector quantity having magnitude equal to its area and direction normal to its surface.
→
If ∆ a j = area vector of jth element and
→
Ej = electric field at jth element ( which can be
considered constant as the area vector is
very small ),
→. →
then electric flux associated with jth element = E j ∆ a j
and the total flux linked with the entire surface is
→ → → →
φ = lim ∑ E j ⋅ ∆a j = ∫ E ⋅ da
→ j
l ∆aj l→0 surface
1.14 Gauss’s Theorem ( or Law )
“The total electric flux associated with any closed surface is equal to the ratio of the total
electric charge enclosed by the surface to ε0.”
→ → 1
∴ φ = ∫ E • da = ∑q
ε0
( Note: ε0 is to be used if the medium in closed surface is vacuum or atmospheric air, else
the permittivity, ε, of the medium has to be used. )
11. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 11
→
The electric field E in the above equation is the resultant electric field due to all charges
whether inside or outside the enclosed surface, but the summation of charges on the RHS of
the above equation is the algebraic sum of the charges enclosed by the surface.
1.15 Application of Gauss’s Theorem
( i ) Electric Field due to an Infinitely Long Straight Charged Wire or Line Charge
Let λ = uniform charge density along the length of
the conductor.
From symmetry, the magnitude of electric field at all
points like P over the curved surface of the cylinder
of radius r and length L, whose axis coincides with
the conductor, will be the same. The direction of the
field at all points on this surface and also at all
points on two ends of the cylinder is radially
outwards if λ > 0.
Using Gauss’s Law,
→ → q λL
∫ E ⋅ da = ∴ 2πrLE =
ε0 ε0
( Surface integration for all points on the two ends of the cylinder will be zero as the field
lines are perpendicular to area vector. )
λ 1 → λ 1 ^ 2k λ ^
∴ E = ⋅ ∴ E = ⋅ r = r
2 π ε0 r 2 π ε0 r r
( ii ) Electric Field due to a Uniformly Charged Infinite Plane Sheet or Sheet of Charge
Let σ = uniform surface charge density on an infinite plane sheet.
P and P’ are points at a perpendicular distance r on either sides of the charged plane. By
symmetry, electric intensity at P and P’ will have equal magnitude and opposite direction. If
the charge on the plane is positive/negative, the direction of the electric intensity will be
away/towards the plane. Consider a closed cylinder with equal lengths on either side of the
plane, from P to P’. As the electric intensity is perpendicular to the plane, the flux linked
with the curved surface of the cylinder is zero. As the points P and P’ are equidistant from
the charged plane, the magnitude of electric intensity are the same.
∴ Ep = Ep’ = E and EpA + Ep’A = 2EA
is the total flux coming out of the cross-sectional
area, A, of the cylinder. The closed cylindrical
surface encloses the charge q = σA.
Using Gauss’s Law,
→ → q σA
∫ E ⋅ da = ∴ 2EA = and
ε0 ε0
12. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 12
σ
E =
2ε 0
The above equation shows that electric intensity at any point is independent of its distance
from the plane.
If two uniformly charged infinite plates,
having surface charge density σ1 and σ2,
are kept parallel to each other, then the
magnitudes and directions of electric
intensity at points between and on either
sides of planes will be as shown in the
figure.
If σ1 = - σ 2 = σ, then the electric intensity
between the plates will be σ / ε0 and on
either sides of the two plates will be zero.
( iii ) Electric Field due to a Uniformly Charged Thin Spherical Shell
Let σ = uniform surface charge density on a spherical shell of radius R.
( a ) For points inside the shell:
Since the charge enclosed in a spherical surface of radius r < R is
zero, the electric intensity is zero at all points inside it.
( b ) For points outside the shell:
Applying Gauss’s Theorem to a spherical surface of radius r > R,
4π R σ
2
4 πr E( r ) =
2
ε0
2
σ R2 q R q 1
∴ E( r ) = = = ,
ε0 r2
2
4 πR ε r
2 4 π ε0 r 2
0
where q is the total charge on the spherical shell. Thus for points outside the spherical shell,
the entire charge of the spherical shell can be treated as concentrated at its centre.
( iv ) Electric Field due to a Uniformly Charged Sphere
Let ρ = uniform volume charge density on a sphere of radius R.
( a ) For points inside the sphere:
Applying Gauss’s Theorem to a sphere of radius r ≤ R,
4π r
3
ρ ρr r
4 πr E( r ) =
2
∴ E( r ) = = E( R )
3 ε0 3ε 0 R
13. 1 - ELECTRIC CHARGE AND ELECTRIC FIELD Page 13
The direction of the field is radially outwards if ρ > 0 and inwards if ρ < 0.
( b ) For points outside the sphere:
Applying Gauss’s Theorem to a sphere of radius r, concentric with charged sphere of radius
R ( r > R ),
4π R
3
ρ Q
4 πr E( r ) =
2
= , where Q is the charge on the sphere.
3 ε0 ε0
3
R ρ Q 1
∴ E( r ) = = (r > R)
3r
2
ε0 4 πε r
2
0
Thus, for points outside the sphere, the entire charge of the sphere can be treated as
concentrated at its centre.