# E&M

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PHYS10342

Electricity & Magnetism

Matthew Evans

A set of notes based primarily on the first-year undergraduate lecture course PHYS10342

at the University of Manchester delivered by Prof. M. Seymour in 2012.

Contents

1

Electrostatics

2

1.1

The Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Coulomb's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

The Electric Field & Superposition Principle . . . . . . . . . . . . . .

2

1.1.3

Continuous Charge Distributions . . . . . . . . . . . . . . . . . . . . .

3

1.2

Gauss's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

Field Lines & Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Quantitative Formulation of Gauss's Law . . . . . . . . . . . . . . . .

4

1.2.3

Divergence of E

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.4

Applications of Gauss's Law . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Conductors & Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1

0c

1

Electrostatics

1.1

The Electric Field

1.1.1

Coulomb's Law

The force, F on a point test charge Q at a position p due to a stationary point charge q at a

position x both charges existing in a vacuum, is given by Coulomb's law:

Qq

F =

r,

(1.1)

40r2

where 0 = 8.854 * 10-12 C2N-1m-2, the permittivity of free space and r2 = (p - x)2, the

displacement vector from the source to the test charge squared. The unit vector r runs parallel

to r. The sign of this force is intuitive, if q and Q have different signs, then F will be negative,

i.e. attractive.

1.1.2

The Electric Field & Superposition Principle

In general, the superposition principle states that, for all linear systems, the response caused by

n stimuli is equal to the sum of each of the responses that would be caused by each individual

stimuli. This idea is going to allow us to simplify the calculation of fields due to given charge

distribution. The force on a test charge, FQ due to a system of charges is the vector sum of all

the forces due to each of the individual charges. For an assembly of n charges,

FQ = FQ + F

+ ... + F

1

Q2

Qn

Q

q1

q2

qn

=

rQ +

rQ + ... +

rQ

4

1

2

n

0

r2

r2

r2

Q1

Q2

Qn

We use the test charge Q to measure the electric field strength at certain points. Hence, E can be

defined as the force per unit test charge. E is a vector field, a single-valued function of spatial

and temporal positions.

FQ

E =

(1.2)

Q

Thus, the electric field due to a point charge, q, is given by,

1

q

E =

r.

40 r2

As this obeys the superposition principle, this expression can be extended for continuous charge

distributions.

2

1.2

Gauss's Law

0c

1.1.3

Continuous Charge Distributions

More often than not, systems are not made up of point charges and instead are spread out

continuously over some region. If this is the case, the sum of field contributions by each charge

becomes an integral over an infinite number of pieces of the region, i.e.

1

1

E(p) =

r dq.

40

r2

If the charge is spread out along a line, L, with a uniform charge density of per unit length,

then dq = dl, where dl is the infinitesimal length element along the line, then the integral

becomes,

1

(x)

E(p) =

r dl,

40

r2

L

where r = p - x, the displacement vector from x to p.

For a surface, S, with a charge per unit area of , then dq = da, where da is the element

area of the surface,

1

(x)

E(p) =

r da.

40

r2

S

Finally, for a volume, V , of charge density per unit volume, dq = dV , where dV is the

volume element,

1

(x)

E(p) =

r dV .

(1.3)

40

r2

V

As equation 1.3 is the most general and realistic case of a charge distribution, it itself is

often referred to as Coulomb's law, instead of equation 1.1.

1.2

Gauss's Law

1.2.1

Field Lines & Flux

To represent a vector field pictorially, we use so-called field lines. The direction of each field

line at a certain point indicates the direction of the field at that point. The strength of the field at

a certain point is represented by the density of field lines drawn. Field lines begin at the sources

of the field and flow out, ending at the sinks, i.e. they flow from positive to negative charges.

Field lines can be useful for getting a "feel" for a field.

3

1.3

Electric Potential

0c

Flux is a measure of the number of fields lines passing through a surface. So, for a surface,

S, with da being an infinitesimal vector area on that surface with magnitude equaling the area

of the surface and direction perpendicular to the surface (in an arbitrary direction for an open

surface, or outwards for a closed surface), the flux, is defined as,

E

E * da.

(2.1)

S

The dot product picks out the component of E along the direction of da. If there are no

sources or sinks (equivalent to same number of sources and sinks) inside a closed surface, then

intuitively = 0; there are as many field lines flowing into the surface as there are out. Fields

lines that originate on a positive charge must either pass out of the surface or end at a negative

charge inside the surface. A charge outside the surface will not make any contribution to the

total flux as and field lines originating outside the surface will pass in one side and out of the

other. This is the crux of Gauss's law, the flux can be used to calculate the amount of charge

enclosed by any arbitrary surface.

We can choose a set of field lines that define some solid angle and argue that the flux of

the field that crosses any area that covers that solid angle is independent of how far away the

surface is or how the surface is oriented. This implies that the total flux through any closed

surface containing a point charge is independent of the size and shape of that surface.

1.2.2

Quantitative Formulation of Gauss's Law

1.2.3

Divergence of E

1.2.4

Applications of Gauss's Law

1.3

Electric Potential

1.4

Conductors & Capacitors

4

Electricity & Magnetism

Matthew Evans

A set of notes based primarily on the first-year undergraduate lecture course PHYS10342

at the University of Manchester delivered by Prof. M. Seymour in 2012.

Contents

1

Electrostatics

2

1.1

The Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Coulomb's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

The Electric Field & Superposition Principle . . . . . . . . . . . . . .

2

1.1.3

Continuous Charge Distributions . . . . . . . . . . . . . . . . . . . . .

3

1.2

Gauss's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

Field Lines & Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Quantitative Formulation of Gauss's Law . . . . . . . . . . . . . . . .

4

1.2.3

Divergence of E

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.4

Applications of Gauss's Law . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Conductors & Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1

0c

1

Electrostatics

1.1

The Electric Field

1.1.1

Coulomb's Law

The force, F on a point test charge Q at a position p due to a stationary point charge q at a

position x both charges existing in a vacuum, is given by Coulomb's law:

F =

r,

(1.1)

40r2

where 0 = 8.854 * 10-12 C2N-1m-2, the permittivity of free space and r2 = (p - x)2, the

displacement vector from the source to the test charge squared. The unit vector r runs parallel

to r. The sign of this force is intuitive, if q and Q have different signs, then F will be negative,

i.e. attractive.

1.1.2

The Electric Field & Superposition Principle

In general, the superposition principle states that, for all linear systems, the response caused by

n stimuli is equal to the sum of each of the responses that would be caused by each individual

stimuli. This idea is going to allow us to simplify the calculation of fields due to given charge

distribution. The force on a test charge, FQ due to a system of charges is the vector sum of all

the forces due to each of the individual charges. For an assembly of n charges,

FQ = FQ + F

+ ... + F

1

Q2

Qn

Q

q1

q2

qn

=

rQ +

rQ + ... +

rQ

4

1

2

n

0

r2

r2

r2

Q1

Q2

Qn

We use the test charge Q to measure the electric field strength at certain points. Hence, E can be

defined as the force per unit test charge. E is a vector field, a single-valued function of spatial

and temporal positions.

FQ

E =

(1.2)

Q

Thus, the electric field due to a point charge, q, is given by,

1

q

E =

r.

40 r2

As this obeys the superposition principle, this expression can be extended for continuous charge

distributions.

2

1.2

Gauss's Law

0c

1.1.3

Continuous Charge Distributions

More often than not, systems are not made up of point charges and instead are spread out

continuously over some region. If this is the case, the sum of field contributions by each charge

becomes an integral over an infinite number of pieces of the region, i.e.

1

1

E(p) =

r dq.

40

r2

If the charge is spread out along a line, L, with a uniform charge density of per unit length,

then dq = dl, where dl is the infinitesimal length element along the line, then the integral

becomes,

1

(x)

E(p) =

r dl,

40

r2

L

where r = p - x, the displacement vector from x to p.

For a surface, S, with a charge per unit area of , then dq = da, where da is the element

area of the surface,

1

(x)

E(p) =

r da.

40

r2

S

Finally, for a volume, V , of charge density per unit volume, dq = dV , where dV is the

volume element,

1

(x)

E(p) =

r dV .

(1.3)

40

r2

V

As equation 1.3 is the most general and realistic case of a charge distribution, it itself is

often referred to as Coulomb's law, instead of equation 1.1.

1.2

Gauss's Law

1.2.1

Field Lines & Flux

To represent a vector field pictorially, we use so-called field lines. The direction of each field

line at a certain point indicates the direction of the field at that point. The strength of the field at

a certain point is represented by the density of field lines drawn. Field lines begin at the sources

of the field and flow out, ending at the sinks, i.e. they flow from positive to negative charges.

Field lines can be useful for getting a "feel" for a field.

3

1.3

Electric Potential

0c

Flux is a measure of the number of fields lines passing through a surface. So, for a surface,

S, with da being an infinitesimal vector area on that surface with magnitude equaling the area

of the surface and direction perpendicular to the surface (in an arbitrary direction for an open

surface, or outwards for a closed surface), the flux, is defined as,

E

E * da.

(2.1)

S

The dot product picks out the component of E along the direction of da. If there are no

sources or sinks (equivalent to same number of sources and sinks) inside a closed surface, then

intuitively = 0; there are as many field lines flowing into the surface as there are out. Fields

lines that originate on a positive charge must either pass out of the surface or end at a negative

charge inside the surface. A charge outside the surface will not make any contribution to the

total flux as and field lines originating outside the surface will pass in one side and out of the

other. This is the crux of Gauss's law, the flux can be used to calculate the amount of charge

enclosed by any arbitrary surface.

We can choose a set of field lines that define some solid angle and argue that the flux of

the field that crosses any area that covers that solid angle is independent of how far away the

surface is or how the surface is oriented. This implies that the total flux through any closed

surface containing a point charge is independent of the size and shape of that surface.

1.2.2

Quantitative Formulation of Gauss's Law

1.2.3

Divergence of E

1.2.4

Applications of Gauss's Law

1.3

Electric Potential

1.4

Conductors & Capacitors

4

# Document Outline

- Electrostatics
- The Electric Field
- Coulomb's Law
- The Electric Field & Superposition Principle
- Continuous Charge Distributions

- Gauss's Law
- Field Lines & Flux
- Quantitative Formulation of Gauss's Law
- Divergence of E
- Applications of Gauss's Law

- Electric Potential
- Conductors & Capacitors

- The Electric Field