Introduction to Electrostatics. The Coulomb's law and Gauss's law.
1 view 0 purchase
Course
B.S. Physics 2023
Institution
Mahatma Gandhi University Kottayam
This document gives a clear introduction to Coulomb's and Gauss's laws. This document is suitable for any graduate course in basic electrostatics. Different types of charge distributions are discussed. The application of Gauss' law to calculate electric fields is discussed. Numerous problems with s...
Introduction to Electrostatics. The Coulomb’s law
and the Gauss’s law
1 The Coulomb’s law
The force between two charges, Q1 and Q2 , is
• Along the line joining them.
• Directly proportional to the product of the charges.
• Inversely proportional to the square of the distance between them.
i.e.,
Q1 Q2
F =k (1)
R2
where Q1 and Q2 are the charges and R is the distance between them. In SI units, the
charge is expressed in Coulombs, R is expressed in meters and force is expressed in Newtons,
1
so that the proportionality constant k = 4πε0
. The quantity ε0 is called the permittivity of free
space. It has the units Farads/meter.
Consider two point charges Q1 and Q2 as shown in the figure. The position vector of Q1 is
r1 and the position vector of Q2 is r2 .
The force on the charge Q2 due to the charge Q1 is
Q1 Q2
F12 = R̂12 (5)
4πε0 R2
where R12 = r2 − r1 is the displacement vector from Q1 to Q2 .
R12
R = |R12 |, and R̂12 = |R12 |
is a unit vector along R12 .
In terms of R12 , we can write the force as
Q1 Q2
F12 = R12 (6)
4πε0 R3
or, in terms of position vectors
Q1 Q2 (r2 − r1 )
F12 = (7)
4πε0 |r2 − r1 |3
Similarly, the force on Q1 due to the charge Q2 is
Q1 Q2 (r1 − r2 ) Q1 Q2 (r1 − r2 )
F21 = = (8)
4πε0 |r1 − r2 |3 4πε0 |r2 − r1 |3
Evidently,
F21 = −F12 (9)
The electric force between two charges exhibits the following characteristics
• Unlike charges attract each other and, like charges repel.
• Two charges act as point charges if their distance from one another is significantly larger
than their spatial dimensions.
• The above-discussed Coulomb law is applicable to static charges, or charges that are at
rest.
2
,1.1 The Principle of Superposition
Assume that a region has a lot of charges. The force that a new charge experiences in this area
is the total of the forces exerted by each of the existing charges on it.
For example, if the charges are Q1 , Q2 ,........,Qn at points specified by position vectors r1 ,
r2 ,........,rn respectively, then the force experienced by a charge Q at a position r is
Q Q1 (r − r1 ) Q Q2 (r − r2 ) Q Qn (r − rn )
F= 3
+ 3
+ ............ +
4πε0 |r − r1 | 4πε0 |r − r2 | 4πε0 |r − rn |3
(10)
Q n Q j (r − r j )
= ∑ |r − r j |3
4πε0 j=1
This is called the superposition principle.
2 Electric field
We know that the effect of a charge is experienced in the region surrounding it. For example,
rubber rods charged by rubbing, attracts tiny pieces of paper, plastic bags rubbed on hair can
deflect the drops of water tripping from a pipe etc.
The region surrounding a charge where its effect is experienced is the electric field. It is
evident that the effect produced by the charge on the surroundings depends on the amount of
charge. To quantify the effect produced by charge, a quantity called electric field intensity is
defined. It is defined as the force per unit charge experienced by a test charge placed in an
electric field. The test charge can be any charge whose magnitude is sufficiently small, so that
it does not affect the background electric field. It should be positive. The direction of electric
field intensity is the direction in which the test charge will move if it was let to do so.
Thus, we have the electric field intensity (or simply electric field)
F
E = lim (11)
Q→0 Q
or simply
F
E= (12)
Q
It is a vector quantity and has the unit Newton/Coulomb (N/C) or Volts/meter (V/m).
The electric field at a point with position vector r due to a point charge at r0 is
Q Q(r − r0 )
E= âR = (13)
4πε0 R2 4πε0 |r − r0 |2
If there are n point charges Q1 , Q2 ,..........,Qn at positions r1 , r2 ,.........,rn receptively, then,
the resultant electric field intensity is
, 2.1 Electric field of Continuous charge distributions
Continuous distributions are characterized by charge densities. These are
• Line charge density ρL (C/m).
• Surface charge density ρS (C/m2 ).
• Volume charge density ρV (C/m3 ).
Figure 2
The electric charge carried by a segment dL of a linear charged material with linear charge
density ρL is
dQ = ρL dL (15)
The total charge is given by
Z
Q= ρL dL (16)
L
The charge carried by an area dS of charged surface with surface charge density ρS is
dQ = ρS dS (17)
The total charge is given by
Z
Q= ρS dS (18)
S
The charge carried by a volume dV of a charged object with a volume charge density ρV is
dQ = ρV dV (19)
The total charge is given by
Z
Q= ρV dV (20)
V
4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller blessonjs. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.99. You're not tied to anything after your purchase.
