Ch 21

21 Electric Charge and Electric Field

Electric ChargeConductors, Insulators, and Induced ChargesCoulomb’s LawElectric Field and Electric ForcesElectric Field CalculationsElectric Field LinesElectric Dipoles

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Objectives1. Apply the concepts of the dichotomy, conservation and quantization of electric charge

2. Given the initial/final charge distribution, calculate the final/initial charge distribution using conservation principles

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Two positive charges or two negative charges repel each other.

A positive charge and a negative charge attract each other.

+ +

+ –

––

+

–

positive charge

negative chargeBenjamin Franklin

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21Electric Charge and the Structure of Matter

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21Electric charge is conserved and quantized.

1. The algebraic sum of all the electric charges in any closed system is constant.

2. Every observable amount of electric charge is always an integer multiple of the electron/proton charge. We say that charge is quantized.

charging process

+ – + – + – – +e. g. rubbing

where n is an integer

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Objectives1. Predict charge distributions, and the resulting attraction or repulsion, in a system of charged insulators and conductors

2. Outline the process of charging

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Conductors permit the easy movement of charge through them, while insulators do not.

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Charging by Induction

In a metallic conductor the mobile charges are always negative electrons.In ionic solutions and ionized gases, both positive and negative charges are mobile.

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21Electric Forces on Uncharged Objects

Polarization is the slight shifting of charge within the molecules of the neutral insulator

Hence a charged object of either sign exerts an attractive force on an uncharged object. A CHARGED body can exert forces on objects that are UNCHARGED.

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21Example:Two neutral conducting pop cans are touching each other. A positively charged balloon is brought near one of the cans as shown below. The cans are separated while the balloon is nearby, as shown. After the balloon is removed the cans are brought back together. When touching again, can X is ____.

A. positively charged B. negatively charged

C. neutral D. impossible to tell13


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A negatively charged balloon is brought near a neutral conducting sphere as shown below. As it approaches, charge within the sphere will distribute itself in a very specific manner. Which one of the diagrams below properly depicts the distribution of charge in the sphere?

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Example:


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A positively charged balloon is brought near a neutral conducting sphere as shown below. While the balloon is near, the sphere is touched (grounded).

At this point, there is a movement of electrons. Electrons move ____ .

A. into the sphere from the ground (hand) B. out of the sphere into the ground (hand)C. into the sphere from the balloonD. out of the sphere into the balloon

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Example:


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Consider three identical conducting cubes, A, B, and C as shown. Cube A and Cube B were made to touch each other, after electrostatic equilibrium is reached, they were separated. Cube A was then made to touch Cube C. After electrostatic equilibrium is reached, what are the final charges on each cube?

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Example:


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You have two lightweight metal spheres, each hanging from an insulating nylon thread. One of the spheres has a net negative charge, while the other sphere has no net charge.

1. If the spheres are close together but do not touch, will they A. attract each other, B. repel each other, or C. exert no force on each other?

2. You now allow the two spheres to touch. Once they have touched, will the two spheres

A. attract each other, B. repel each other, or C. exert no force on each other?

Exercise 1:

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Objective1. Calculate the net electric force on a point charge exerted by a system of point charges

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21Charles Augustin de Coulomb (1736–1806) Coulomb’s Law

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21: Force between two point chargesCoulomb’s Law

Where:

SI unit of electric charge : coulomb (C)SI unit of force: newton (N)SI unit of length : meter (m)

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q1 q2 +x–axis

+y–axis

a

What are the magnitude and direction of the force exerted by q1 on q2? By q2 on q1?

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Example:


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q1 q2 +x–axis

+y–axis

a

What are the magnitude and direction of the force exerted by q1 on q2? By q2 on q1?

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Example:


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q1 q2 +x–axis

+y–axis

a

What are the magnitude and direction of the net force exerted by q2 and q3 on q1?By q1 and q2 on q3?

a

q3

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Example:


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Five equal charges Q are equally spaced on a semicircle of radius R as shown. Find the force on a charge q located at the center of the semicircle.

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Example:


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When two electrically charged particles in empty space interact, how does each one know that the other is there? What goes on in the space between them to communicate the effect of each one to the other?

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Objectives1. Describe the electric field due to a point charge quantitatively and qualitatively

2. Establish the relationship between the electric field and the electric force on a test charge

3. Predict the trajectory of a massive point charge in a uniform electric field

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Electric Field

The electric field is the intermediarythrough which A communicates its presence to B (qo).

The electric force on a charged body is exerted by the electric field created by other charged bodies.

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21Electric Field

electric force Fo experienced by a test charge qo at a certain point, divided by the charge qo

SI unit of electric field magnitude : N/C

(for point charge only) 28


21Electric Field of a Point Charge

(magnitude only)

(magnitude + direction) 29

𝑬 = 𝒌𝒒𝒓𝟐

𝑬 = 𝒌𝒒𝒓𝟐 𝒓'


21What are the magnitude and direction of the net electric field at point P?

a

P

q1 q2 +x–axis

+y–axis

aa

q3

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Example:If a test charge qo is placed at point P, what are the magnitude and direction of the electric force it experiences?


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A square has equal negative charges at three of its corners, as shown. The direction of the electric field at point P is

P1

2

3

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Example:


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A proton is moving horizontally to the right in an electric field that points vertically upward. The electric force on the proton is

A. zero.B. upward.C. downward.D. to the left.E. to the right.

q = +e+

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Example:

An electron is moving horizontally to the right in an electric field that points vertically upward. The electric force on the proton is

A. zero.B. upward.C. downward.D. to the left.E. to the right.


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q = –e–

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Example:


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1. A negative point charge moves along a straight-line path directly toward a stationary positive point charge. Which aspect/s of the electric force on the negative point charge will remain constant as it moves?

A. magnitudeB. directionC. both magnitude and directionD. neither magnitude nor direction

2. A negative point charge moves along a circular orbit around a positive point charge. Which aspect/s of the electric force on the negative point charge will remain constant as it moves?

A. magnitudeB. directionC. both magnitude and directionD. neither magnitude nor direction 34

Example:


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Objective1. Evaluate the electric field at a point in space due to a system of arbitrary charge distributions

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21Superposition of Electric Fields

Components:

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𝐸 = ?


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Point charge Charge distribution

q Qr

PP

At point P,

(simple) (complicated)

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𝑬 = 𝒌𝒒𝒓𝟐𝒓'


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Q

P

q1

q2q3

qn

qo

(Superposition of forces)

(Superposition of E–fields)38


21Charge distributions • linear charge distribution (1-D)λ (Greek letter lambda): linear charge density (charge per unit length. measured in C/m)

Examples:

• surface charge distribution (2-D)σ (Greek letter sigma): surface charge density (charge per unit area: measured in C/m2)

• volume charge distribution (3-D)ρ (Greek letter rho): volume charge density (charge per unit volume: measured in C/m3)

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𝑬 =𝟐𝒌 +𝝀𝒙 =

𝟐𝒌𝝀𝒙

𝑬 =𝟐𝒌 +𝝀𝒙 =

𝟐𝒌𝝀𝒙


21Line chargeFor an infinite line of charge:

+𝝀

Px

At point P:Magnitude:

Direction: +x-direction

At point R:Magnitude:

Direction: –x-direction

x

y

R x

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For an infinite line of charge:

−𝝀





x

y

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Line charge

𝑬 =𝟐𝒌 −𝝀𝒙 =

𝟐𝒌𝝀𝒙

𝑬 =𝟐𝒌 −𝝀𝒙 =

𝟐𝒌𝝀𝒙

PxR x


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𝑬 =+𝝈𝟐𝝐𝟎

=𝝈𝟐𝝐𝟎

𝑬 =+𝝈𝟐𝝐𝟎

=𝝈𝟐𝝐𝟎


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+𝝈

Px





x

y

R x

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For an infinite plane sheet of charge:

Plane charge


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−𝝈





x

y

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𝑬 =−𝝈𝟐𝝐𝟎

=𝝈𝟐𝝐𝟎

𝑬 =−𝝈𝟐𝝐𝟎

=𝝈𝟐𝝐𝟎

PxR x

For an infinite plane sheet of charge:

Plane charge


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Consider three charged infinite planes that are parallel to each other asshown in the figure. The rightmost plane has uniform charge density +σwhile the rest have uniform charge density –σ. Calculate the magnitude ofthe electric field at the four regions I to IV.

–σ –σ +σ

I II III IV

L L

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Example:


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An infinite line of charge with linear charge density λ was placed a distanced on top of an infinite sheet with surface charge density σ as shown. Whatis the net electric field at point P at a distance of 2d from the line?

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Example:


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Three infinite sheets of uniform surface charge densities of +σ, +σ and –σ are arranged as shown. Which of the regions 1, 2, 3 and 4 has/have netelectric fields pointing in the +x-direction?

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Exercise 2:


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Objective1. Given the electric field lines, deduce the electric field vectors and nature of electric field sources

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• Field lines never intersect


21An electric field line is an imaginary line or curve drawn through a region of space.

• Its tangent at any point is in the direction of the electric–field vector at that point

• It is not the same as trajectories.

At any particular point, the electric field has aunique direction.

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Suppose the electric field lines in a region of space are straight lines. If a charged particle is released from rest in that region, will the trajectory of the particle be along a field line?

A. YESB. NO

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Example:


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Given the charges +Q and –2Q, which of the following diagram shows a correct representation of the electric field lines?

B.

C.

D.

A.

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Example:


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Objective1. Discuss the motion of an electric dipole in a uniform electric field

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21What is an Electric Dipole?

• pair of point charges with equal magnitude and opposite sign • (a positive charge +q and a negative charge –q) separated by a distance d

: electric dipole moment: directed from (–) to (+)

+ –+q –q

d

p = qd (magnitude)SI units of p : C • m 56


21Force and Torque on an Electric Dipole

In a uniform external electric field :

:

:

Magnitude of torque:

Direction of torque:

X (into the page)

Torque

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(Magnitude of torque)with

(torque in vector form)

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Force and Torque on an Electric Dipole


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21Potential Energy of an Electric Dipole

(potential energy of a dipole)

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Ch 21