Physics II (PH2223)

Physics for Scientists and

Engineers, with Modern Physics, 4th edition, Giancoli

Topics Covered

!! Electric Charge & Electric Field

!! Electric Potential

!! Capacitance, Dielectric, Electric Energy Storage

!! Electric Currents & Resistance

!! DC Circuits

!! Magnetism

!! Sources of Magnetic Field

!! Electromagnetic Induction & Faraday’s Law

!! Inductance, Electromagnetic Oscillations, and AC Circuits

Electric Charge &

Electric Field

"! Electric charge is a property of matter.

"! Charge is either positive or negative; attractive forces between unlike charges and repulsive forces between like charges.

"! Charge is conserved (symmetry of nature).

"! Charge is quantized (there are no free charged particles with charge less than that of an electron).

"! Charge is a multiple of the electron or proton charge:

e = 1.602 !10"19C

Electric Charge

"! Ordinary matter is made up of atoms which have positively charged nuclei (protons and neutrons) and negatively charged electrons surrounding them.

"! Conductors:

#! Solid: one or more of the bound electrons can move freely in material.

#! Some solids or liquids: ions are able to move about and give rise to currents (electrolytes).

#! Gasses: neutral atoms or molecules become ionized (external).

"! Insulators or dielectrics: Electric charges do not freely move.

"! Semiconductors: some bound charges are able to move (by thermal vibration, light, electric field,…).

"! The Earth: Can be considered to be an infinite reservoir of (or for) electrons, i.e., it can accept or supply an unlimited number of electrons.

Matter

Electrostatics

Electrostatics deals with the phenomena and properties of stationary or slow-moving charges, which involves the buildup of charge on the surface of objects.

Induction:

A negatively charged object repels electrons from the surface of a second object.

Two electric charges experience electrostatics force between them such that:

#! It is proportional to the square of the separation r between the 2 particles along the line that joins them

#! It is proportional to the product of the magnitude of the charges, q and Q, on the two particles.

#! It is attractive if the charges are of opposite sign and repulsive if the charges have the same sign.

Coulomb's Law

! !" #

$

!

!"!

!#!

!F = k qQ

r2r

k = 14!"0

! 9 "109 N #m2

C2

$%&

'()

Problem (1)

Two balls, one with charge Q1 = +Q and the other with charge Q2= +2Q, are held at a separation d. Is there any place between Q1 and Q2 where the

force on any charge (+ or -) is zero?

Electric Field

"! Test charge does not exert force on other charges, hence does not disturb the charges in the vicinity. In practical situations, a test charge can be approximated

by a charge of nearly negligible magnitude.

"! Electric field exists at any point in space where a test charge, if placed at that

point would experience an electric force. The electric field is a vector quantity.

The direction of the electric field at a point is the same as direction of the force

experienced by a positive test charge placed at that point.

!Fg = m0

!g !F0 = q0

!E similar to

Electric Field Lines An electric charge emits an electric field which always points towards a negative charge and points away from a positive charge:

Note that only a few of the possible field lines are drawn and the field is continuous and exists between the lines.

Electric Field Lines "! A system of charged particles:

#! The lines must begin on positive charges and must terminate on negative charges

#! The number of lines leaving a positive charge or approaching a negative charge is proportional to the magnitude of the charge.

#! No two field lines can cross each other.

"! Drawing electric field lines:

#! The electric field vector, E, is tangent to the electric field lines at each point.

#! The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the E-field in a given region.

Problem (2)

Based on the Figure to the right, which of the following is correct?

A.! q1 = q2

B.! q2 is positive and q1 negative; |q1| > |q2|

C.! q1 is positive and q2 negative; |q1| > |q2|

D.! q2 is positive and q1 negative; |q2| > |q1|

Electric Field of a Point Charge

"! If a small test charge q0 is placed at a distance r from charge q, it will experience a force (Coulomb force)

"! If a point charge q0 is placed at a position where the electric field is E, then the force on q0 is

"! Comparing these two expressions for F, we see that

F = 14!"0

qq0r2

F = q0E

E = 14!"0

qr2

Problem (3) An alpha particle (charge +2e) is sent at high speed toward a gold nucleus (charge +79e). What is the electrical force acting on the alpha particle when it is

2.0 ! 10!14 m from the gold nucleus? Is the Coulomb force a repulsion force or

an attraction force?

Superposition Principal

If more than two charged particles exist in a system, then the Coulomb force exerted on one particle is the summation of all of the Coulomb forces between that particle and the rest of the particles in the ensemble:

Similarly, the electric field E at a point due to several charges is the vector sum of the fields due to the individual charges.

!E = 1

4!"0qiri2 ri

i#

!F = 1

4!"0qiq0ri2 ri

i# = Fi x

i

N

# + Fi yi

N

# + Fizi

N

#

For a continuous charge distribution we can write:

!E = 1

4!"0dqr2r#

Electric Field: System of Charges

When summing E-fields from multiple charges, it is best to draw a vector diagram first at a point where the E-field is to be determined. Draw the E-field vector of each charge in the direction as dictated by

Then vectorially add the individual E-fields together, letting the diagram determine the sign of the E-field. Then add using the absolute value of the charge:

The + or - is selected based upon the direction of the E-field vector.

!E = 1

4!"0qr2r.

E = ±( )1

n

! 14"#0

qr2.

Problem (4)

Positive charges are situated at three corners of a rectangle as shown in the figure below. Find the electric field at the fourth corner.

The simplest kind of charge distribution is a point charge. If the dimensions of the charge distribution are much less than the distance away from a point of interest, it can be considered a point charge.

If the dimensions must be considered, then the charge density can be useful.

Volume density Area density Linear density

Charge Distribution

In the special case for uniform charge density, we have:

! = dqdV

C m3( ) ! = dqdA

C m2( ) ! = dqdl

C m( )

! = qV

C m3( ) ! = qAC m2( ) ! = q

lC m( )

Problem (5)

A rod of length 2a has a uniform positive charge per unit length, ! and a total charge Q. Calculate the electric field at a point P, a distance x along the perpendicular bisector of the rod.

Electric Dipole An electric dipole is an important case of a two-charge distribution, which consists of a pair of point charges with equal magnitude and opposite sign, +Q and –Q, separated by a fixed distance d.

Let's take the magnitude of !r+ and !r! to be equal to r,!r+ = r sin" x ! r cos" y

r+ =!r+r= sin" x ! cos" y

!E+ =

14#$0

Q+

r2 r+ =1

4#$0

Q+

r2 sin" x ! cos" y( )

!E! =

14#$0

Q!

r2 r! =1

4#$0

Q!

r2 !sin" x ! cos" y( )!Etotal =

!E+ +

!E!

= 14#$0

Qr2 sin" x ! cos" y( ) +

14#$0

Qr2 !sin" x ! cos" y( )

!Etotal = ! 1

4"#0

Qr22cos$( ) y

An important property of an electric dipole is the torque exerted on it by a uniform electric field. The torque is given by

The product of the charge q and the separation d is called electric dipole moment, denoted by p. Electric dipole moment, p, is a vector quantity and its direction is defined from –q to +q.

Torque & Force on an Electric Dipole

! = d 2( )qE sin" + d 2( )qE sin"! = dqE sin"

!! = !r "!F

!F = q

!E

#$%

&%'!! = !r " q

!E

U = ! !p "

!E

!! = !p "

!E

Potential energy of a dipole in an electric field is:

Similar to projectile motion (from physics I) and Newton’s Second Law:

Motion of Charged Particles in a Uniform Electric Field

F = qE = ma ! a = qEm

ax = 0 and ay =(!e)Em

t = 0 "x0 = y0 = 0v0 x = v0 and v0 y = 0

#$%

t > 0 "x = v0t

y = 12ayt

2 = y = ! 12eEmt2

#$&

%&

y = ! 12eEmv0

2 x2