Download Electric Charges and Fields

Lecture 2 : Electric charges and fields
• Electric charge
• Electric field
• Introducing Gauss’s Law
Electric charge
• Electric charge is an intrinsic property of the
particles which make up matter, which
experimentally can be either positive or negative
Electric charge
• Electric charge is quantized such that protons and
electrons have equal and opposite charge 𝒆 =
± 𝟏. 𝟔 × 𝟏𝟎−𝟏𝟗 𝑪 [unit 𝐶 = Coulombs]
Electric charge
• Electric charge is locally conserved and cannot be
created or destroyed
Electric charge
• Two electric charges attract or repel each other with
equal and opposite forces
Electric charge
• The strength of the force is given by Coulomb’s Law
𝒒𝟏 𝒒𝟐
𝑭=
𝒓
𝟐
𝟒𝝅𝜺𝟎 𝒓
𝑞1
𝑟
𝑞2
𝐹
• The force is proportional to the
magnitude of the charges 𝑞1 , 𝑞2
𝐹
• The force is inversely proportional to
the square of the separation 𝑟
• The force acts along the line joining
the charges : by symmetry, no other
direction could be singled out
• The force strength is governed by
the permittivity of free space 𝜀0
1
where
= 9 × 109 𝑁𝑚2 𝐶 −2
4𝜋𝜀0
Vectors
• A vector is a quantity – such as a force – which has both a
magnitude and a direction
• It can be indicated by 𝐹 or F or F (helpful!)
• A vector can be specified by its components along coordinate axes, such as 𝐹 = (𝐹𝑥 , 𝐹𝑦 , 𝐹𝑧 )
• The magnitude of a vector is 𝐹 =
𝐹𝑥 2 + 𝐹𝑦 2 + 𝐹𝑧 2
• A unit vector, indicated by 𝐹, has magnitude = 1
• You may also see 𝐹 = 𝐹𝑥 𝑖 + 𝐹𝑦 𝑗 + 𝐹𝑧 𝑘, in terms of unit
vectors along co-ordinate axes
Vectors
• Vectors may be added by summing their components
• The dot product of two vectors, 𝑎. 𝑏, is the projection of one
vector along the other, 𝑎 𝑏 cos 𝜃 (𝜃 = angle between). It
can also be evaluated as 𝑎. 𝑏 = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧
• The cross product of two vectors, 𝑎 × 𝑏, is a vector
perpendicular to both with magnitude 𝑎 𝑏 sin 𝜃
Electric charge
• The total force from multiple charges is given by the
principle of superposition
𝐹13
𝐹12
𝑞1 𝑞2
𝑞1 𝑞3
=
𝑟12 +
𝑟13
2
2
4𝜋𝜀0 𝑟12
4𝜋𝜀0 𝑟13
𝑞1
𝑟12
𝐹1 = 𝐹12 + 𝐹13
𝑟13
𝑞2
𝑞3
Electric field
• How is a force communicated between charges? A
useful model is the electric field
• We interpret that electric charges set up an electric
field 𝐸 in the region of space around them
• Then, a “test charge” 𝒒, placed in the electric field,
will feel a force 𝑭 = 𝒒𝑬
• The electric field is a vector “force field” representing
the size/direction of the force per unit charge
Electric field
• The electric field can be represented by field lines:
• Electric field lines start on positive charges and end on negative charges.
Their direction shows the force acting on a test charge; their spacing
indicates the force strength. They can never cross.
Electric field
• A more complicated example of an electric field:
There are no electric field lines here, because
a test charge experiences zero net force
Electric field
• The electric field around a point charge +𝑸 follows
simply from Coulomb’s Law
𝑞
𝑟
• Place a test charge +𝑞 at
distance 𝑟 from +𝑄
• Force 𝐹 =
𝑄𝑞
𝑟
4𝜋𝜀0 𝑟 2
• Electric field 𝑬 =
𝑭
𝒒
=
𝑸
𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐
Introducing Gauss’s Law
• For general distributions of charges, it would be a nightmare
to work out the electric field through the superposition
𝑞𝑖
principle 𝐸 = 𝑖
2 𝑟𝑖
4𝜋𝜀0 𝑟𝑖
• Luckily we can use a more powerful method, Gauss’s Law.
This is equivalent to Coulomb’s Law, but more convenient
• Gauss’s Law says that for any closed surface 𝑺, the flux of the
electric field 𝑬 through 𝑺 is equal to the total charge
enclosed by 𝑺, divided by 𝜺𝟎
• In mathematical terms:
𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
𝐸 . 𝑑𝐴 =
𝜀0
Surface integrals
• We are familiar with the concept of integration as summing
𝑏
up the “area under a curve” – written as 𝑎 𝑓 𝑥 𝑑𝑥
Surface integrals
• This can be generalized to more dimensions! A 2D integral
sums up the volume under a surface, 𝑓 𝑥, 𝑦 𝑑𝑥 𝑑𝑦
Surface integrals
• The flux over a surface sums up the projection (dot product)
of a vector field across the surface
• First, we break the surface into area elements 𝑑 𝐴 (these are
vectors, in the direction normal to the surface)
𝑑𝐴
surface
area element 𝑑𝐴
Surface integrals
• For each area element, we evaluate the dot product with the
vector field, 𝐸. 𝑑𝐴 = 𝐸 𝑑𝐴 cos 𝜃 (𝜃 = angle between)
𝑑𝐴
𝐸
𝑑𝐴
This case has
zero flux!
𝐸
• We sum this up over the surface to obtain the “flux” 𝐸. 𝑑 𝐴
• Important : we will only ever consider cases where 𝑬 is
perpendicular to the surface, so 𝑬. 𝒅𝑨 = 𝑬 × 𝑨𝒓𝒆𝒂
Introducing Gauss’s Law
𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
𝐸 . 𝑑𝐴 =
𝜀0
𝑆
• Breaking it down …
• 𝑆 is any closed surface (i.e., with no
edges/gaps)
• Consider a small area element ∆𝐴 of 𝑆,
where the vector is normal to 𝑆
• The flux of 𝐸 through the area element
is equal to 𝐸. ∆𝐴 = 𝐸 ∆𝐴 cos 𝜃
• The surface integral 𝐸. 𝑑𝐴 means the
total flux of 𝐸 through the surface
• Gauss’s Law says that this is equal to
the total charge enclosed divided by 𝜀0
Introducing Gauss’s Law
𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
𝐸 . 𝑑𝐴 =
𝜀0
• Why is it the same as Coulomb’s Law?
Let’s derive Gauss’s Law …
• From the definition of solid angle we
∆𝐴 cos 𝜃
𝑟2
𝑞
𝐸=
4𝜋𝜀0 𝑟 2
know that ∆Ω =
𝑆
• Coulomb’s law :
• Flux through ∆𝐴 is 𝐸. ∆𝐴 =
𝐸 ∆𝐴 cos 𝜃 =
𝑞 ∆𝐴 cos 𝜃
4𝜋𝜀0 𝑟 2
=
𝑞 ∆Ω
4𝜋𝜀0
• Integrating over the whole surface:
𝑞
𝑞
𝐸. 𝑑𝐴 =
𝑑Ω =
4𝜋𝜀0
𝜀0
• This holds true for any surface 𝑆
Introducing Gauss’s Law
Example: point charge +𝑄
𝑟
𝑆
𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
𝐸 . 𝑑𝐴 =
𝜀0
• By spherical symmetry, the
electric field 𝐸 is radial
• Choose a spherical surface 𝑆 of
radius 𝑟 centred on the charge
• 𝐸 has the same magnitude
across the surface and cuts it at
right angles, so 𝐸. 𝑑 𝐴 = 𝐸 × 𝐴
• Gauss’s Law then simplifies to:
𝑄
𝐸 × 4𝜋𝑟 2 =
𝜀0
𝑄
• Re-arranging: 𝐸 =
4𝜋𝜀0 𝑟 2
Summary
• Electric charge is a fundamental property of nature
• Electric charges 𝑞1 , 𝑞2 at a distance 𝑟 attract/repel
𝑞1 𝑞2
with a Coulomb force 𝐹 =
2𝑟
4𝜋𝜀0 𝑟
• Electric charges set up an electric field 𝐸 in the
region of space around them; a test charge 𝑞 placed
in the field feels a force 𝐹 = 𝑞𝐸
• A powerful method for deriving the electric field is
𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑
Gauss’s Law for a closed surface, 𝐸. 𝑑 𝐴 =
𝜀0