Physics / Higher Physics 1A

Electricity and Magnetism

Revision

Electric Charges

Two kinds of electric charges

Called positive and negative

Like charges repel

Unlike charges attract

Coulomb’s Law

In vector form,

is a unit vector

directed from q1 to q2

Like charges produce a

repulsive force between

them

F12

ke

q1q

2

r2

ˆ r

ˆ r

The Superposition Principle

The resultant force on q1 is the vector

sum of all the forces exerted on it by

other charges: F1 = F21 + F31 + F41 + …

Electric Field

Continuous charge distribution

2ˆe

e

o

qk

q r

FE r

2 20

ˆ ˆl imi

i

e i eq

i i

q d qk k

r r

E r r

Electric Field Lines – Dipole

The charges are

equal and opposite

The number of field

lines leaving the

positive charge

equals the number

of lines terminating

on the negative

charge

Electric Flux

0

s u r fa c e

l imi

E i iA

E A d

E A

E E

i A

icos

i E

i A

i

Gauss’s Law

qin is the net charge inside the surface

E represents the electric field at any point on the

surface

E E d A

qin

0

Field Due to a Plane of Charge

The total charge in the surface is σA

Applying Gauss’s law

Field uniform everywhere

E 2 EA

A

0

, and E

2 0

Properties of a Conductor in

Electrostatic Equilibrium

1. Electric field is zero everywhere inside conductor

2. Charge resides on its surface of isolated conductor

3. Electric field just outside a charged conductor is perpendicular to the surface with magnitude σ/εo

4. On an irregularly shaped conductor surface charge density is greatest where radius of curvature is smallest

Work done by electric field is F.ds = qoE.ds

Potential energy of the charge-field system is

changed by ΔU = -qoE.ds

For a finite displacement of the charge from A

to B, the change in potential energy is

Electric Potential Energy

B

B A oA

U U U q d E s

Electric Potential, V

The potential energy per unit charge, U/qo, is the electric potential

The work performed on the charge is W = ΔU = q ΔV

In a uniform field

B

A

o

UV d

q

E s

B B

B AA A

V V V d E d E d E s s

Equipotential Surface

Any surface consisting

of a continuous

distribution of points

having the same

electric potential

For a point charge

e

qV k

r

From V = -E.ds = -Exdx

Along an equipotential surfaces V = 0 Hence E ds

i.e. an equipotential surface is perpendicular to the electric field lines passing through it

Finding E From V

x

d VE

d x

V Due to a Charged

Conductor

E · ds = 0

So, potential difference

between A and B is zero

Electric field is zero inside

the conductor

So, electric potential

constant everywhere inside

conductor and equal to

value at the surface

Cavity in a Conductor

Assume an irregularly

shaped cavity is inside a

conductor

Assume no charges are

inside the cavity

The electric field inside the

conductor must be zero

Definition of Capacitance

The capacitance, C, is ratio of the charge on either conductor to the potential difference between the conductors

A measure of the ability to store charge

The SI unit of capacitance is the farad (F)

QC

V

Capacitance – Parallel Plates

Charge density σ = Q/A

Electric field E = /0 (for conductor) Uniform between plates, zero elsewhere

C Q

V

Q

Ed

Q

Q

0A

d

0A

d

Capacitors in Parallel

Capacitors can be replaced with one capacitor with a capacitance of Ceq

Ceq = C1 + C2

Capacitors in Series

Potential differences add up to the battery voltage

Q Q1 Q

2

V V1 V

2

V

Q

V1

Q1

V

2

Q2

1

C

1

C1

1

C2

Energy of Capacitor

Work done in charging the capacitor appears as electric potential energy U:

Energy is stored in the electric field

Energy density (energy per unit volume)

uE = U/Vol. = ½ oE2

2

21 1( )

2 2 2

QU Q V C V

C

Capacitors with Dielectrics

A dielectric is a nonconducting material that, when placed between the plates of a capacitor, increases the capacitance

For a parallel-plate capacitor

C = κCo = κεo(A/d)

Rewiring charged capacitors

Two capacitors, C1 & C2 charged to same potential difference, Vi.

Capacitors removed from battery and plates connected with opposite polarity.

Switches S1 & S2 then closed. What is final potential difference, Vf?

Q1i, Q2i before; Q1f, Q2f after.

Q1i = C1Vi; Q2i = -C2Vi

So Q=Q1i+Q2i=(C1-C2)Vi

But Q= Q1f+Q2f (charge conserved)

With Q1f = C1Vf; Q2f = C2Vf hence Q1f = C1/C2 Q2f

So, Q=(C1/C2+1) Q2f

With some algebra, find Q1f = QC1/(C1+C2) & Q2f = QC2/(C1+C2)

So V1f = Q1f / C1 = Q / (C1+C2) & V2f = Q2f / C2 = Q / (C1+C2)

i.e. V1f = V2f = Vf, as expected

So Vf = (C1 - C2) / (C1 + C2) Vi, on substituting for Q

Magnetic Poles

Every magnet has two poles

Called north and south poles

Poles exert forces on one another

Like poles repel

N-N or S-S

Unlike poles attract

N-S

Magnetic Field Lines for a Bar Magnet

Compass can be used to trace the field lines

The lines outside the magnet point from the North pole to the South pole

Direction

FB perpendicular to plane formed by v & B

Oppositely directed forces are exerted on charges of different signs

cause the particles to move in opposite directions

Direction given by Right-Hand Rule

Fingers point in the direction of v

(for positive charge; opposite direction if negative)

Curl fingers in the direction of B

Then thumb points in the direction of v x B; i.e. the direction of FB

The Magnitude of F

The magnitude of the magnetic force on a charged particle is FB = |q| vB sin

is the angle between v and B

FB is zero when v and B are parallel

FB is a maximum when perpendicular

Force on a Wire

F = I L x B

L is a vector that points in the direction of the current (i.e. of vD)

Magnitude is the length L of the segment

I is the current = nqAvD

B is the magnetic field

Force on a Wire of Arbitrary Shape

The force exerted segment ds is

F = I ds x B

The total force is

Ib

d a

F s B

Force on Charged Particle

Equating the magnetic & centripetal forces:

Solving gives r = mv/qB

F qvB mv

2

r

Biot-Savart Law

dB is the field created by the current in the length segment ds

Sum up contributions from all current elements I.ds

B

0

4

I d s ˆ r

r2

B for a Long, Straight Conductor

B

0I

2 a

B for a Long, Straight Conductor, Direction

Magnetic field lines are circles concentric with the wire

Field lines lie in planes perpendicular to to wire

Magnitude of B is constant on any circle of radius a

The right-hand rule for determining the direction of B is shown Grasp wire with thumb in direction of

current. Fingers wrap in direction of B.

Magnetic Force Between Two Parallel Conductors

Parallel conductors carrying currents in the same direction attract each other

Parallel conductors carrying currents in opposite directions repel each other

F1

0I

1I

2

2 al

Definition of the Ampere

The force between two parallel wires can be used to define the ampere

When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

F1

l

0I

1I

2

2 a with

0 4 10

7 T m A

-1

Ampere’s Law

The line integral of B . ds around any closed path equals oI, where I is the

total steady current passing through any surface bounded by the closed path.

B ds 0I

Field in interior of a Solenoid

Apply Ampere’s law

The side of length ℓ inside the

solenoid contributes to the field

Path 1 in the diagram

BdsBdd1path 1path

sBsB

B 0

N

l

I 0nI

Ampere’s vs. Gauss’s Law

Integrals around closed path vs. closed surface. i.e. 2D vs. 3D geometrical figures

Integrals related to fundamental constant x source of the field.

Concept of “Flux” – the flow of field lines through a surface.

B ds 0I

E dA q

0

Gauss’ Law in Magnetism

Magnetic fields do not begin or end at any point i.e. they form closed loops, with the number of lines

entering a surface equaling the number of lines leaving that surface

Gauss’ law in magnetism says:

B B .d A 0

Faraday’s Law of Induction

The emf induced in a circuit is directly proportional to the rate of change of the magnetic flux through that circuit

Nd

B

dt

QuickTime™ and a

Cinepak decompressor

are needed to see this picture.

Ways of Inducing an emf

Magnitude of B can change with time

Area enclosed, A, can change with time

Angle can change with time

Any combination of the above can occur

d

dt

BA cos

Motional emf

Motional emf induced in a conductor moving through a constant magnetic field

Electrons in conductor experience a force, FB = qv x B that is directed along ℓ

In equilibrium, qE = qvB or E = vB

Sliding Conducting Bar

Magnetic flux is

The induced emf is

Thus the current is

d

B

dt

d

dt

Blx Bldx

dt

Blv

B Blx

I

R

Blv

R

Induced emf & Electric Fields

A changing magnetic flux induces an emf and a current in a conducting loop

An electric field is created in a conductor by a changing magnetic flux

Faraday’s law can be written in a general form:

Not an electrostatic field because the line integral of E.ds is not zero.

E .ds d

B

dt

Generators

Electric generators take in energy by work and transfer it out by electrical transmission

The AC generator consists of a loop of wire rotated by some external means in a magnetic field

Rotating Loop

Assume a loop with N turns, all of the same area, rotating in a magnetic field

The flux through one loop at any time t is:

B = BA cos = BA cos wt

Nd

B

dt

NABd

dt

cos w t NAB w sin w t

Motors

Motors are devices into which energy is transferred by electrical transmission while energy is transferred out by work

A motor is a generator operating in reverse

A current is supplied to the coil by a battery and the torque acting on the current-carrying coil causes it to rotate

Eddy Currents

Circulating currents called eddy currents are induced in bulk pieces of metal moving through a magnetic field

From Lenz’s law, their direction is to oppose the change that causes them.

The eddy currents are in opposite directions as the plate enters or leaves the field

Equations for Self-Inductance

Induced emf proportional to the rate of change of the current

L is a constant of proportionality called the inductance of the coil.

L L

dI

dt

Inductance of a Solenoid

Uniformly wound solenoid having N turns and length ℓ. Then we have:

B 0nI

0

N

l

I

B BA

0

NA

l

I

L N

B

I

0N

2

A

l

Energy in a Magnetic Field

Rate at which the energy is stored is

Magnetic energy density, uB, is

II

d U dL

d t d t

U L IdI

0

I

12LI

2

uBU

Al

B

2

2 0

RL Circuit

Time constant, tL / R, for the circuit

t is the time required for current to reach 63.2% of its max value

I R

1 e Rt

L

R1 e

tt