Coulomb's Law and Electric Field Intensity: A Comprehensive Guide with Exercises, Exercises of Electromagnetism and Electromagnetic Fields Theory

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Chapter 2

Coulomb’s Law and Electric Field Intensity

 (^) In 1600, Dr. Gilbert, a physician from England, published the first major classification of electric and non-electric materials.  (^) He stated that glass, sulfur, amber, and some other materials “not only draw to themselves straw, and chaff, but all metals, wood, leaves, stone, earths, even water and oil.”  (^) In the following century, a French Army Engineer, Colonel Charles Coulomb, performed an elaborate series of experiments using devices invented by himself.  (^) Coulomb could determine quantitatively the force exerted between two objects, each having a static charge of electricity.  (^) He wrote seven important treatises on electric and magnetism, developed a theory of attraction and repulsion between bodies of the opposite and the same electrical charge.

The Experimental Law of Coulomb

The Experimental Law of Coulomb

 (^) The permittivity of free space ε is measured in farads per meter (F/m), and has the magnitude of: 12 9 0

8.854 10 10 F m

   

 (^) The Coulomb’s law is now: 1 2 2 0

Q Q

F

 R

 (^) The force F acts along the line joining the two charges. It is repulsive if the charges are alike in sign and attractive if the are of opposite sign.

The Experimental Law of Coulomb

1 2 2 2 12 0 12

Q Q

 R

F  a

 (^) In vector form, Coulomb’s law is written as:  (^) F 2 is the force on Q 2 , for the case where Q 1 and Q 2 have the same sign, while a 12 is the unit vector in the direction of R 12 , the line segment from Q 1 to Q 2

12 12 12  R a R 12 12 R  R (^) 2 1 2 1    r r r r

F = 1?

1 2 1 2 2 21 0 12 1 4 Q Q  R F =  F  a

The Experimental Law of Coulomb

 (^) Example A charge Q 1

• C at M (1,2,3) and a charge of Q 2

• C at N (2,0,5) are located in a vacuum. Determine the force exerted on Q 2 by Q 1

R 12 (^)  r 2 (^)  r 1 (2 0 5 ) (1 2 3 ) x y z x y z  a  a  a  a  a  a 1 2 2 x y z  a  a  a 12 12 12  R a R 1 (1 2 2 ) 3 x y z  a  a  a 1 2 2 2 12 0 12 1 4 Q Q  R F  a 4 4 12 2 1 (3 10 )( 10 ) 1 (1 2 2 ) 4 (8.854 10 ) 3 3 x y z           a a a 10 20 20 N x y z  a  a  a

Electric Field Intensity

 (^) Let us consider one charge, say Q 1 , fixed in position in space.  (^) Now, imagine that we can introduce a second charge, Q t , as a “unit test charge”, that we can move around.  (^) We know that there exists everywhere a force on this second charge ► This second charge is displaying the existence of a force field. 1 2 1 0 1

t t t t

Q Q

 R

F  a

 (^) The force on it is given by Coulomb’s law as:  (^) Writing this force as a “force per unit charge” gives: 1 2 1 0 1

t t t t

Q

Q  R

F

a Vector Field,

Electric Field Intensity

Electric Field Intensity

 (^) For a charge which is not at the origin of the coordinate, the electric field intensity is: 2 0

Q



^ 

  ^ 

r r

E r =

r r r^ r

3 0

Q



^ 

^ 

r r

r r

3 2 2 2 2 0

1 (^ )^ (^ )^ (^ )

x y z

Q x x y y z z

x x y y z z



a a a

Electric Field Intensity

 (^) The electric field intensity due to two point charges, say Q 1 at r 1 and Q 2 at r 2 , is the sum of the electric field intensity on Q t caused by Q 1 and Q 2 acting alone (Superposition Principle). 2 1 (^0 ) 2 2 (^0 )

Q

Q

 

E r = a

r r

a

r r

Field Due to a Continuous Volume Charge Distribution  (^) We denote the volume charge density by ρ v , having the units of coulombs per cubic meter (C/m 3 ).  (^) The small amount of charge Δ Q in a small volume Δ v is v  Q    v  (^) We may define ρ v mathematically by using a limit on the above equation: 0

lim

v v

Q

v

  

 (^) The total charge within some finite volume is obtained by integrating throughout that volume: vol v Q   dv 

Field Due to a Continuous Volume Charge Distribution  (^) Example Find the total charge inside the volume indicated by ρv = 4 xyz 2 , 0 ≤ ρ ≤ 2, 0 ≤ Φ ≤ π /2, 0 ≤ z ≤ 3. All values are in SI units. x   cos y   sin 2  v (^)   4  sin   cos z vol Q  (^)   vdv 3 2 2 2 0 0 0 (4 sin cos )( ) z z d d dz                       3 22 3 2 0 0 0 4 z sin cos d d dz         3 2 2 0 0 16 z sin cos d dz       sin 2  2sin  cos 3 2 0  8 z dz  72 C

Field of a Line Charge

 (^) Now we consider a filamentlike distribution of volume charge density. It is convenient to treat the charge as a line charge of density ρ L C/m.  (^) Let us assume a straight-line charge extending along the z axis in a cylindrical coordinate system from –∞ to +∞.  (^) We desire the electric field intensity E at any point resulting from a uniform line charge density ρ L

E

z z

d d dE

 

E  a  a

Field of a Line Charge

 (^) The incremental field dE only has the components in a ρ and a z direction, and no a Φ direction. • Why?  (^) The component dE z is the result of symmetrical contributions of line segments above and below the observation point P.  (^) Since the length is infinity, they are canceling each other ► dEz = 0.  (^) The component dE ρ exists, and from the Coulomb’s law we know that dE ρ will be inversely proportional to the distance to the line charge, ρ.

E

z z

d d dE

 

E  a  a

Field of a Line Charge

 (^) Now let us analyze the answer itself: 0

L    

E  a

 (^) The field falls off inversely with the distance to the charged line, as compared with the point charge, where the field decreased with the square of the distance.

Field of a Line Charge

 (^) Example D2.5. Infinite uniform line charges of 5 nC/m lie along the (positive and negative) x and y axes in free space. Find E at: ( a ) P A (0,0,4); ( b ) P B

PA PB 0 0 ( ) 2 2 x x y L Ly A x y P         E  a  a 9 9 0 0 5 10 5 10 2 (4) 2 (4) z z        a  a 44.939 V m z  a 0 0 ( ) 2 2 x x y L (^) Ly B x y P         E  a  a 9 9 0 0 5 10 5 10 (0.6 0.8 ) 2 (5) 2 (4) y z z        a  a  a 10.785 36.850 V m y z  a  a

• (^) ρ is the shortest distance between an observation point and the line charge