E&M

Text-only Preview

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

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