electricity & magnetism (fall 2011) lecture # 4 by moeen ghiyas
TRANSCRIPT
ELECTRICITY & MAGNETISM (Fall 2011)
LECTURE # 4
BY
MOEEN GHIYAS
TODAY’S LESSON
(Electric Field & Gauss Law)
Fundamentals of Physics by Halliday / Resnick / Walker (Ch 23 / 24)
Today’s Lesson Contents
• Electric Field lines
• Motion of Charged Particles in Uniform Electric Field
• Electric Flux
Quiz # 1
• Time : 15 min
Electric Field Lines
• A convenient way of visualizing electric field patterns
is to draw lines that follow the same direction as the
electric field vector at any point. These lines, called
electric field lines, are related to the electric field in
any region of space in the following manner:
– The electric field vector E is tangent to the electric field line at
each point.
– The number of lines per unit area through a surface
perpendicular to the lines is proportional to the magnitude of
the electric field in that region. Thus, E is great when the field
lines are close together and small when they are far apart.
Electric Field Lines
• Figure – Non uniform Electric Field
– Electric field is more intense on surface A than on surface B
– The lines at different locations point in different directions
indicates that the field is non-uniform.
Electric Field Lines
• Figure - Electric field lines for the field due to a single charge
– The lines are actually directed radially from the charge in all
directions; thus, instead of the flat “wheel” of lines shown in
2 dimension, you should picture an entire sphere of lines
Electric Field Lines
• Figure - Electric field lines for the field due to a single charge
– Why lines are directed outward for +ve charge and inward for -ve?
– Because a positive test charge placed in this field would be
repelled radially away from the positive point charge while same
would be attracted toward the negative charge
Electric Field Lines
• Figure - Electric field lines for the field due to a single charge
– In either case of +ve or –ve charge, the lines are along the radial
direction and extend all the way to infinity.
– Note that the lines become closer together as they approach the
charge; this indicates that the strength of the field increases as
we move toward the source charge.
Electric Field Lines
• The rules for drawing electric field lines are:
– The lines must begin on a positive charge and terminate on a
negative charge.
– The number of lines drawn leaving a positive charge or
approaching a negative charge is proportional to the magnitude
of the charge.
– No two field lines can cross.
Electric Field Lines
• Is this visualization of the electric field in terms of field
lines consistent with Coulomb’s law?
• To answer this question, consider an imaginary spherical
surface of radius r concentric with a point charge.
• From symmetry, we see that the magnitude of the electric
field is the same everywhere on the surface of the
sphere.
• The number of lines N that emerge from the charge is
equal to the number that penetrate the spherical surface.
Electric Field Lines
• The number of lines N that emerge from the charge is equal to
the number that penetrate the spherical surface.
• Hence, the number of lines per unit area on the sphere is
N/4πr 2 (where the surface area of the sphere is 4πr 2).
• Because 4π is constant, E is proportional to the number of
lines per unit area, i.e., E varies as 1/r 2; this finding is
consistent with Coulomb’s law equation.
• Also for two charges, if object 1 has charge Q1 and object 2
has charge Q2, then ratio of number of lines is N2 / N1 = Q2 / Q1
Electric Field Lines
• Problems with model of qualitatively describing the
electric field with the lines?
1. We draw a finite number of lines from (or to) each
charge, which appear as if the field acts only in certain
directions at certain places; instead, the field is
continuous—that is, it exists at every point.
2. Danger of gaining the wrong impression from a two-
dimensional drawing of field lines being used to describe
a three-dimensional situation.
Electric Field Lines
• The lines are nearly radial at points close to either
charge,
• The same number of lines emerge from each charge
because the charges are equal in magnitude.
• At great distances from the charges, the field is
approximately equal to that of a single point charge of
magnitude 2q (q+q = 2q).
Electric Field Lines
• Now we sketch the electric field lines associated with
a positive charge +2q and a negative charge -q.
• In this case, the number of lines leaving +2q is twice
the number terminating at -q.
• Hence, only half of the lines that leave the positive
charge reach the negative charge.
• The remaining half terminate on a negative charge
we assume to be at infinity.
• At distances that are much greater than the charge
separation, the electric field lines are equivalent to
those of a single charge q (i.e. 2q-q = q).
Motion of Charged Particles in Uniform Electric Field
• When a particle of charge q and mass m is placed in an
electric field E, We know E = F / q
• Then electric force on the charge is , F = E q
• But, from Newton’s 2nd law we know F = m a
• Thus, we have Fe = E q = m a
• The acceleration of the particle is then a = E q / m
• If E is uniform (i.e. constant in magnitude and direction),
then the acceleration ‘a’ is constant.
• If q is +ve, then its acceleration is in the direction of the
electric field, and opposite for negatively charged particle
Motion of Charged Particles in Uniform Electric Field
• The electric field in the
region between two
oppositely charged flat
metallic plates is
approximately uniform
• Suppose an electron of
charge -e is projected
horizontally into this field
with an initial velocity vi i.
Motion of Charged Particles in Uniform Electric Field
• Because the electric field E in figure is
in the +y direction, the acceleration of
the electron is in the -y direction i.e.
• Since acceleration is constant, we can
apply the equations of kinematics in
two dimensions with vxi = vi and vyi = 0.
• After the time t, the components of its
velocity are
Motion of Charged Particles in Uniform Electric Field
• Its coordinates after time t are
• Substituting the value t = x / vi from
eqn 1 into eqn 2, we see that y is
proportional to x2. Hence, the trajectory
is a parabola.
• After the electron leaves the field, it
continues to move in a straight line in
the direction of v, obeying Newton’s
first law, with a speed v > vi
Motion of Charged Particles in Uniform Electric Field
• Since we are dealing with atomic
particles, note that we have neglected
the gravitational force acting on the
electron.
• For an electric field of 104 N/C, the
ratio of the magnitude of the electric
force eE to the magnitude of the
gravitational force mg is of the order of
1014 for an electron and of the order of
1011 for a proton.
Motion of Charged Particles in Uniform Electric Field
• Example – An Accelerated Electron – An electron enters the
region of a uniform electric field as shown in fig, with v i =
3.00 x 106 m/s and E = 200 N/C. The horizontal length of the
plates is l = 0.100 m.
(a) Find the acceleration of the electron while it is in the
electric field.
Solution: e = 1.60 x 10-19 C,
m = 9.11 x 10-31 kg
Motion of Charged Particles in Uniform Electric Field
• Example – An Accelerated Electron – An electron enters the
region of a uniform electric field as shown in fig, with v i =
3.00 x 106 m/s and E = 200 N/C. The horizontal length of the
plates is l = 0.100 m.
(b) Find the time it takes the electron to travel through the
field.
Solution:
Electric Flux – (Uniform Electric Field)
• Electric field lines - qualitative method
• Electric flux - quantitative way
• Consider an electric field that is uniform in
both magnitude and direction as shown
• The total number of lines penetrating the
surface is proportional to the product EA
• Product of the magnitude of the electric field
E and surface area A perpendicular to the
field is called the electric flux ΦE : ΦE = EA
• From SI units flux is measured as Nm2/C
Electric Flux• Example – What is the electric flux through a sphere
that has a radius of 1.00 m and carries a charge of
+1.00 μC at its centre?
• Solution:
• Area of sphere = 4πr2 = 12.6 m2
• Note: The field points radially outward and is therefore
everywhere perpendicular to the surface
Electric Flux
• What if the surface through which electric field passing
is not perpendicular to the field
Electric Flux
• Flux through a surface of fixed area A has
– A max value EA when the surface is perpendicular to the field
(when the normal to the surface is parallel to the field (θ = 00);
– the flux is zero when the surface is parallel to the field (when
the normal to the surface is perpendicular to the field (θ = 900);
Electric Flux – (Non Uniform Electric Field)
• If the electric field varies (non-uniform) over a given
surface and generally it does. Then, our definition of flux
is applied only over a small element of area.
• Consider a general surface divided up into a large
number of small elements, each of area ΔA. The
variation in the electric field over one element can be
neglected if the element is sufficiently small.
Electric Flux
• For vector ΔAi magnitude is the area of the ith element of
the surface and direction is perpendicular to the surface
element, as shown. The electric flux ΔΦE through this
element is
• By summing the contributions of all elements, we obtain
the total flux through the surface.
Electric Flux
• If we let the area of each element approach zero, then
the general definition of electric flux is
• Above equation is a surface integral, which means it
must be evaluated over the surface in question.
Electric Flux
• We are often interested in evaluating the flux through a
closed surface, that divides space into an inside and an
outside region
• Consider a closed
surface as in figure
Electric Flux• The net flux through the surface is proportional to the
net number of lines leaving the surface, where the net
number means the number leaving the surface minus
the number entering the surface.
• If more lines are leaving than entering, the net flux is
positive.
• If more lines are entering than leaving, the net flux is
negative.
Electric Flux• Using the symbol to represent an integral over a
closed surface, we can write the net flux ΦE as
• where En represents the component of the electric field
normal to the surface
Electric Flux• Example – Consider a uniform electric field E oriented
in the x direction. Find the net electric flux through the
surface of a cube of edges ℓ , oriented as shown.
Electric Flux• Solution: The net flux is the sum of the fluxes through
all faces of the cube.
• First, note that flux through four of the faces ( , , and
the unnumbered ones) is zero because E is perpendicular
to dA on these faces.
• Net flux through faces and is
33 44
11 22
Summary / Conclusion
• Electric Field Lines
• Motion of Charged Particles in Uniform Electric Field
• Electric Flux