Semester 2 Physics (SF 026)yslphysics.weebly.com/uploads/4/8/2/6/48261583/slides_1.pdf · (2D, maximum four charges) ... 2. b. the electric force that q 2 exerts on q 1. ... particle

Semester 2

Physics

(SF 026)

Lecture: BP 3

by Yew Sze Ling @ Fiona

Website:

http://yslphysics.weebly.com/

Chapter 1: Electrostatics The study of electric charges at rest, the forces

between them and the electric fields associated with

them.

Overview Electrostatic

Coulomb’s Law Electric Field Electric

Potential

2r

QqF

Electric Field For

Point Charge

q

FE

2r

kQE

Charge in a Uniform

Electric Field

d

VE

oq

WV

Electric

Potential Energy

2r

kQqU

Equipotential

Surface

1.1 Coulomb’s Law

State Coulomb’s law,

Sketch the electric force diagram and

apply Coulomb’s law for a system of point

charges. (2D, maximum four charges)

224 r

kQq

r

QqF

o

Learning Objectives

Coulomb’s Law

Coulomb’s law states that the magnitude of the

electrostatic (Coulomb/electric) 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.

r

QqF

2

Electrostatic Force

Coulomb’s Law

Equation

Coulomb’s Law

Mathematically,

where

F : magnitude of electrostatic (Coulomb’s) force

Q , q : magnitude of charges

r : distance between two point charges

k : electrostatic constant,

k = 9.0 × 109 N m2 C-2

εo : permittivity of free space,

εo = 8.85× 10-12 C2 N-1 m-2

242 r

Qq

r

kQqF

o

04

1

k

Coulomb’s Law Graphically,

Notes:

oThe sign of the charge can be ignored when

substituting into the Coulomb’s law equation.

oThe sign of the charges is important in distinguishing

the direction of the electric force.

r

F

0

F

2

1

r0

Gradient, m = kQq

Electric Force Diagram

There are two types of charges in nature – positive and negative charges.

Like charges repel – Repulsive force

Unlike charges attract – Attractive force

The direction of the force is along the straight line joining the two point charges.

Example 1

Sketch the force diagram for q1.

a)

b)

+ − −

+

− +

q1

q1

q2 q3

q2

q3

F21

F31

F21

F31

Example 2

Two point charges, q1 = −20 nC and q2 = 90 nC, are

separated by a distance of 4.0 cm as shown in figure

below.

Find the magnitude and direction of

a. the electric force that q1exerts on q2.

b. the electric force that q2 exerts on q1.

(Given Coulomb’s constant, k = 9.0×109 N m2 C-2)

Example 2 – Solution


Example 3

Three point charges lie along the x-axis as shown in figure

below.

Calculate the magnitude and direction of the total electric

force exerted on q2.





Example 4

Figure below shows the three point charges are placed in

the shape of triangular.

Determine the magnitude and direction of the resultant

electric force exerted on q1.





Example 5

Two point charges, q1 = +4.0µC and q2 = +6.0µC, are

separated by a distance of 50 cm as shown in figure

below.

Determine the position of a point charge q that is placed

on the line joining q1 and q2 such that the net force acting

on it is zero.



1.2 Electric Field

Define and use electric field strength,

Use for point charge.

Sketch the electric field strength diagram and determine electric field strength E for a system of charges. (2D, maximum four charges)

q

FE

2r

kQE

Learning Objectives

Electric Field

Electric field is defined as a region of space around isolated charge where an electric force is experienced if a (positive) test charge is placed in the region.

Electric field around charges can be represented by drawing a series of lines. These lines are called electric field lines (lines of force).

The direction of electric field is tangent to the electric field line at each point.

E

+ +

Electric Field Lines

(a) Isolated point charge

Single positive charge Single negative charge

The lines point radially

outward from the charge

The lines point radially

inward from the charge


(b) Two charges

Two equal point charges of opposite sign, +q and -q

The lines are curved and

they are directed from the

positive charge to the

negative charge.


Two equal positive charges, +q and +q

• Neutral point is defined as a

point (region) where the

total electric force is zero.

• It lies along the vertical

dash line.


Two opposite unequal charges, +2q and -q

• note that twice as many lines

leave +2q as there are lines

entering –q

• number of lines is

proportional to magnitude

of charge.


(c) Two opposite charged parallel metal

plate

• The lines go directly from positive

plate to the negative plate.

• The field lines are parallel and equally

spaced in the central region far from

the edges but fringe outward near the

edges. Thus, in the central region, the

electric field has the same magnitude at

all points.

• The fringing of the field near the edges

can be ignored because the separation

of the plates is small compared to their

size.


Characteristic of electric field lines: • The field lines indicate the direction of the electric field (the

field points in the direction tangent to the field line at any point).

• The lines are drawn so that the magnitude of electric field is proportional to the number of lines crossing unit area perpendicular to the lines. The closer the lines, the stronger the field.

• Electric field lines start on positive charges and end on negative charges, and the number starting or ending is proportional to the magnitude of the charge.

• The field lines never cross because the electric field don’t have two value at the same point.

Example 6

Sketch electric field lines for the diagram below:

a)

b)

Electric Field Strength

The electric field strength at a point is defined as the

electric (electrostatic) force per unit (positive) test

charge.

Mathematically

where,

E : magnitude of the electric field strength

F : magnitude of the electric force

q0 : magnitude of test charge

0q

FE

Electric Field Strength

Since,

thus,

It is a vector quantity.

The units of electric field strength is N C-1 or V m-1.

The direction of the electric field strength, E depends on

the sign of isolated charge.

2r

kQqF

q

r

kQq

E2

2r

kQE

In the calculation

of magnitude E,

substitute the

MAGNITUDE

of the charge

only.

What is the difference between

isolated charge and test charge?

0q

FE

2

r

QkE

Test charge

Isolated charge

E

+

+

Isolated charge

Test charge

Since both F and E are vectors, how

to determine the direction of F and E

for a test charge? ?

• The direction of electric

field strength, E depends on

sign of isolated point charge.

• The direction of the electric

force, F depends on the sign

of isolated point charge and

test charge.

A positive isolated point charge

A negative isolated point charge

Example 7

A metal sphere can be regarded as a point object in space. It

carries a charge of +6.0 µC. Find the electric field that the

sphere generates at a distance of 10 cm around it.

Solution

Example 8 Two point charges, q1= − 1 C and q2= 4 C, are placed 2 cm

and 3 cm from the point A respectively as shown in figure

below.

Find

a) the magnitude and direction of the electric field

intensity at point A.

b) the total electric force exerted on q0 = − 4 C if it is

placed at point A.






Example 9

Two point charges, q1 = − 12 nC and q2 = 12 nC, are placed

0.10 m apart. Calculate the total electric field at point a

and point b.






1.3 Electric Potential

Define electric potential,

Define and sketch equipotential lines and

surfaces of an isolated charge and a uniform

electric field

Use for a point charge and a

system of charges.

oq

WV

r

kQV

Learning Objectives

1.3 Electric Potential

Calculate potential difference between two

points

Deduce the change in potential energy

between two points in electric field

Calculate potential energy of a system of

point charges

o

initialfinalq

WVVV

VqU

23

32

13

31

12

21

r

qq

r

qq

r

qqkU

Learning Objectives

Electric Potential

Electric potential, V of a point in the electric field is

defined as the work done in bringing (positive) test charge

from infinity to that point in the electric field per unit test

charge.

oq

WV W∞ : Work done

qo : Test charge

Electric Potential

OR

Electric potential, V of a point in an electric field is defined as the potential energy per unit positive charge at that point in the electric field.

• It is a scalar quantity.

• The unit of electric potential is Volt (V) OR J C 1.

• The electric potential at infinity is considered zero. (V∞ = 0)

oq

UV U : Potential energy

qo : Test charge

Extra Note

+

r

A • Q

+qo

FE Fext

r = ∞

2 where

r

kQqF

drFdW

drFdW

drFdW

oE

rx

x

E

rx

x

E

ext

r

kQqW

rkQqW

drrkQqW

o

r

o

r

o

1

2

r

kQV

rq

kQqV

q

WV

A

o

oA

o

rA

The electric potential at

point A at distance r from

a positive point charge Q

Electric Potential

Since

thus the equation of electric potential can also be written as

r

kQqW o

r

kQV

Electric Potential

The electric potential energy of a positively charged

particle increases when it moves to a point of higher

potential.

The electric potential energy of a negatively charged

particle increases when it moves to a point of lower

potential.

If the value of work done is negative – work done by

the electric force (system).

If the value of work done is positive –work done by

the external force or on the system.

In the calculation of U, W and V, the sign of the charge

MUST be substituted in the related equations.

U=qoV

Equipotential Surface

Equipotential surface (line)is defined as the locus of

points that have the same electric potential.

A point charge A uniform electric field

Equipotential Surface

• The dashed lines represent the equipotential surface

(line).

• The equipotential surfaces (lines) always perpendicular to

the electric field lines passing through them and points in

the direction of decreasing potential.

VA = VB ≠ VC

• From the figures, then the work done to bring a test

charge from B to A is given by

0

BAo

ABoBA

VVq

VqW No work is done in

moving a charge along

the same equipotential

surface.

Example 10

Figure below shows a point A at distance 10 m from the

positive point charge, q = 5C.

Calculate the electric potential at point A and describe the

meaning of the answer.



Example 11

Two point charges, q1 = +0.3 C and q2 = −0.4 C are

separated by a distance of 6 m as shown in figure below.

Calculate the electric potential at point A if point A is at

the midpoint of q1 and q2.



Example 12

Two point charges, q1 = +12 nC and q2 = −12 nC are

separated by a distance of 8 cm as shown in figure below.

Determine the electric potential at point P.



Potential Difference

0q

WVVV initialfinal OR

o

BABAAB

q

WVVV

Potential Difference The work done to bring a charge from one point to

another in the field does not depend on the path taken

(because the work done by conservative force).

Example 13

Two points, S and T are located around a point charge of

+5.4 nC as shown in figure below.

Calculate

a) the electric potential difference between points S and T.

b) the work done in bringing a charge of 1.5 nC from

point T to point S.




Changes in Potential Energy

From the definition of electric potential difference, V

Therefore the change in a potential energy is given by

0

Δq

WV UW and

0

Δq

UV

VqU Δ0

Ufinal ‒ Uinitial Vfinal ‒ Vinitial

Potential Energy of A System

The total electric potential energy of the system of

charges is the total work done to bring all the charges from

infinity to their final positions.

The total potential energy, U can be expressed as

23

32

13

31

12

21

r

qq

r

qq

r

qqkU

Extra Note

In the system of charges, suppose there were originally no

charges at the points A, B and C as in the figure above.

Step 1:

The charge q1 is brought from infinity and placed at

point A. Since originally there were no charges, the

charge q1 does not experience any electric force when it is

brought from infinity, that is F = 0. Hence VA = 0.

Since U1 = q1VA, hence

1 0U

Extra Note

Step 2:

With the charge q1 fixed at point A, an electric field is

produced by the q1. Electric potential to bring the charge

q2 from infinity to the point B at a distance of r12 from A

is

Since U2 = q2VB, hence

1

12

B

kqV

r

1 22

12

kq qU

r

Extra Note

With the charge q1 fixed at point A and q2 at the point B,

the electric potential at the point C is

Since U3 = q3VC, hence

1 2

13 23

C

kq kqV

r r

1 3 2 33

13 23

kq q kq qU

r r

Extra Note

Therefore, the electric potential energy of the system,

321 UUUU

23

2

13

13

12

210r

q

r

qkq

r

qkq

23

32

13

31

12

21

r

qkq

r

qkq

r

qkq

23

32

13

31

12

21

r

qq

r

qq

r

qqkU

Example 14 Two point charges, Q1= +2.0 C and Q2= 6.0 C, are

placed 4.0 cm and 5.0c m from a point P respectively as

shown in figure below.

a. Calculate the electric potential at P due to the charges.

b. If a charge Q3= +3.0 C moves from infinity to P,

determine the change in electric potential energy for

this charge.

c. When the charge Q3 at point P, calculate the electric

potential energy for the system of charges.





1.4 Charge In A Uniform Electric

Field Explain quantitatively with the aid of a

diagram the motion of charge in a uniform

electric field.

Use for uniform electric field.

Cases:

• Stationary charge

• Charge moving perpendicularly to the field

• Charge moving parallel to the field

• Charge in dynamic equilibrium

d

VE

Learning Objectives

Uniform Electric Field

A uniform electric field is represented by a set of

electric field lines which are straight, parallel to each

other and equally spaced

It can be produced by two flat parallel metal plates which

is charged, one with positive and one is negative and is

separated by a distance.

Direction of E: (+) ve plat to (–) plat

Case 1 : Stationary Charge

Positive stationary charge Negative stationary charge

Force experienced by charge is in the

same direction as electric field, E. Force experienced by charge is in the

opposite direction as electric field,

E.


Consider a stationary particle of charge qo and mass m

is placed in a uniform electric field E, the electric

force Fe exerted on the charge is given by

Since only electric force exerted on the particle, thus

this force contributes the nett force, F and causes the

particle to accelerate.

According to Newton’s second law, then the magnitude

of the acceleration of the particle is

EqF oe

maEq

maFF

o

e

m

Eqa e


Because the electric field is uniform (constant in

magnitude and direction) then the acceleration of the

particle is constant.

If the particle has a positive charge, its acceleration is

in the direction of the electric field. If the particle has a

negative charge (electron), its acceleration is in the

direction opposite the electric field.

Case 2 : Charge moving

perpendicularly to the field

Positive charge Negative charge

• The positive charge will be

deflected and moves along a

parabolic path towards the

negative plate.

• The positive charge moves under

the influence of the electric force

which is at the same direction as

electric field lines.

• The negative charge will be

deflected and moves along a

parabolic path towards the

positive plate.

• The negative charge moves under

the influence of the electric force

which is opposite direction to the

electric field lines.


perpendicularly to the field Consider an electron (e) with mass, me enters a uniform

electric field, E perpendicularly with an initial velocity

u, the upward electric force will cause the electron to

move along a parabolic path towards the upper plate.

The electric force Fe exerted on the charge,

EqF oe


perpendicularly to the field From Newton’s second law,

Therefore the magnitude of the electron’s acceleration is

given by

maF

upward) :(direction e

ym

eEa 0xa;

maEq

maFF

o

e

x -

component

y - component

u

v

s

a

uvx

0u

yyyy

yyy

sauv

tauv

2

22

tus xx

tvus

tatus

yyy

yyy

2

1

2

1 2

0

e

ym

eEa

The path makes by the electron is similar to the motion of a ball projected horizontally above the ground


parallel to the field Fe and v → in the same

direction Fe and v → in the opposite

direction

• The electric force on the positive

charge is in the same direction as to

its motion.

• The positive charge accelerates

along a straight line.

• The electric force on the positive

charge is in the opposite direction

to its motion.

• The positive charge decelerates

along a straight line.

Case 4 : Charge in dynamic

equilibrium

Between electric force and weight

• Particle weight is at the

opposite direction to the

electric force.

• Dynamic equilibrium

means the charge moves

with constant velocity.

• Only particles with this

constant speed can pass

through without being

deflected by the fields.

WFe

Uniform Electric Field

The graph is a straight line with

negative constant gradient, thus

d

VE

d

V

r

VE

0

0

Example 15

Two parallel plates are separated 5.0 mm apart. The

electric field strength between the plates is 1.0 104 N

C1. A small charge of +4.0 nC is moved from one

conducting plate to another. Calculate

a. the potential difference between the plates

b. the work done on the charge.


Example 16

If the plates are horizontal and separated by 1.0 cm,

the plates are connected to 100 V battery, the

magnitude of the electric field is 1.0×104 N C-1. If an

electron is released from the rest at the upper plate,

determine

a. the acceleration of the electron.

b. the speed and the kinetic energy required to travel

to the lower plate.

c. the time required to travel to the lower plate.

[mass of electron = 9.11×10-31 kg;

charge of electron = 1.60×10-19 C ]




Example 17 The figure shows a section of the deflection system of a cathode ray

oscilloscope. An electron travelling at a speed of 1.5×107 m s-1 enters

the space between two parallel metal plates 60 mm long. The electric

field between the plates is 4.0×103 V m-1.

a. Copy the figure, sketch the path of the electron in between plates,

and after emerging from the space between the plates.

b. Find the acceleration of the electron between the plates.

c. Determine the velocity when it emerges from the space between

the plates.



Summary

Electrostatic

Coulomb’s Law Electric Field Electric

Potential

2r

QqF

Electric Field For

Point Charge

q

FE

2r

kQE

Charge in a Uniform

Electric Field

d

VE

oq

WV

Electric

Potential Energy

2r

kQqU

Equipotential

Surface

Documents

Semester 2 Physics (SF 026)yslphysics.weebly.com/uploads/4/8/2/6/48261583/slides_1.pdf · (2D, maximum four charges) ... 2. b. the electric force that q 2 exerts on q 1. ... particle