E-field Lines and Equipotential surfacesIn Figurexa01 below, draw some of the E-field lines between the objects
and identify the sign of the surface charge on the object at point A.
[Equipotential maps around charged objects.]
The E-field lines are perpendicular to the equipotential lines. The
E-field lines are from +15V object (high potential) to the +10V object
(low potential), so between the objects, the +15V object is positively
charged at the surface and +10V object is negatively charged at the
surface. In addition, as both objects carry ne
Charge in a circular coilA charge q is uniformally distributed along a circular coil of radius R.
By taking into account the symmetry of the problem, calculate the
electric field and the electrostatic potential created by this charge in
the point P(0,0,z) located on the axis of the coil.
Find the electric field from the potential previously calculated.
The symmetry of the problem indicates us that the field is directed
along the z axis :
$$\cos \theta = \frac{z}{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{./figures//ScriptR
Study of density functionConsider the volume density of charge ρ(x,y,z)u2004=u2004ρ(x) of the figure on
the left, infinite in the directions y and z, where: aρ₀₊u2004=u2004Au2006[Cu2006m⁻²].
a) Determine and graphically represent the electric field associated for
all of the values of x. Determine and graphically represent the
potential V(x), with the condition V(−a/2)u2004=u20040.
b) Now consider the charge density ρ(x,y,z)u2004=u2004ρ(x) of the figure on the
right side, where bρ⁰⁻u2004=u2004aρ₀₊u2004=u2004A. Determine and graphi
Planar capacitor connected to a batteryA planar capacitor, whose plates are separated by l, is connected to a
battery producing a difference of potential V₀, as indicated in the
figure. A dipole constituted of two masses m of opposite charges u2005±u2005q
and of length 2a is placed between the plates of the capacitor, tilted
of a small angle θ with respect to the lines of electric field. What is
the angular frequency of the oscillations of the dipole when we release
it ? The moment of inertia of the dipole : Iu2004=u20042ma².
The electric fiel
Multipolar expansionif ru2004≫u2004r′, the "multipolar expansion" :
\frac{1}{4\pi\epsilon_0} [\frac{Q}{r}+\frac{\bf{\hat{r}}\cdot \bf{p}}{r^2}+ ...] ~,whereQisthetotalchargeandmomentofthedistribution,isanapproximationforthepotentialatpointP.[Suggestion:consideraTaylorexpansionofBut:so:and:Finally:=4πϵ01[rQ+r2r^⋅p] .
This approach is general. We can also reach this result more directly : Free surface of a vortexConsider the vortex formed by an ideal fluid which flows by a hole at
the bottom of a container. The potential of velocities for this flow is,
with a correct approximation :
Φ(θ)=2πΓθ ,
where Γ is a constant. Check that the flow is irrotational. Then use the
Bernoulli law to determine the shape of the free surface of the vortex.
Considering the symmetry of the problem, we will use the cylindrical
coordinates (r,θ,z). For the given potential the velocity field is:
Infinite, perpendicular and charged platesa) Consider two infinite plates uniformly charged and perpendicular to
the x axis. The first one, at xu2004=u20040 carries a positive charge density
u2005+u2005σ₀; the second one, at xu2004=u2004d, a negative charge density u2005−u2005σ₀.
Calculate the field and the potential.
b) Same questions, but this time the two plates carry the same density
of charge u2005+u2005σ₀.
a) By symmetry the field is E(r)=Ex(x)x^.
For the plate in xu2004=u20040 we have:
Rod on charged railsA rod of mass m, of length l and of resistance R slips without friction
on two parallel rails. A battery imposes a potential difference ℰ
between the rails, and a uniform and constant magnetic field B is
perpendicularly applied on the plane defined by the rails (Fig. 1). We
will neglect the self-inductance of the circuit. Initially, the rod is
at rest. Show that at istant t, the velocity is:
v=Blε(1−e−B2l2t/mR) .
The rod accelerates towards the left under the action of the L Potential of a ring and of a diska) A thin circular ring of radius R has a uniformly distributed charge
Q. Determine the electric potential at a point P on the axis of the ring
a distance x from its center, Fig.xa0[fig:rings] (left) (take that
potential at the infinity is 0: V(∞)u2004=u20040).
b) A thin flat disk, of radius R₀, has a uniformly distributed charge Q,
Fig.xa0[fig:rings] (right). Determine the potential at a point P on the
axis of the disk, a distance x from its center (take that potential at
the infinity is 0: V(∞)u2004=u200
Metallic layer in a planar capacitorThe electrodes of a planar capacitor, of surface A and separated by a
distance d, carry free chargesu2005±u2005Qu2004=u2004u2005±u2005σA. A metallic layer, uncharged
and with thickness a, is inserted between the two electrodes, as in the
figure.
a) Determine the difference of potential and the capacity of the
capacitor. Show that the capacity is equivalent to the one of two
capacitors in series, for whom we will give the characteristics.
b) Show that the capacity of the capacitor is not modified if the
me
Ball hanging on a springA ball of mass m is hanging (vertically) on a spring of elastic constant
k and of natural length l₀. The suspension point of the spring is
attached to the ceiling at height H.
-
Determine the expression of the total potential energy V (n.b.
define carefully the reference of potential energy).
-
Determine the equilibrium position of the ball.
-
The two potential energies that come into play here are the
gravitational potential energy V_(g)(z), which is a function of the
height z, and the
Dielectric in a planar capacitorWe consider a planar capacitor, where d is the separation between the
electrodes. In the absence of dielectric material, the capacity is C₀.
What becomes this capacity when a layer of dielectric of constant ϵ_(r)
and thickness fd (0u2004
Uniformly charged diskLet us consider a flat disk of radius R carrying an uniformally
distributed charge Q. The surface charge density is σ.
Calculate the electric field and the potential in a point P on the axis
of the disk as function of the distance z between P and the disk (Use
the results of exercise 1).
Deduce the electric field created by the plate of infinite dimension
with the same charge density σ (make R tend to infinity).
In the case where R is finite, discuss the limit cases zu2004≫u2004R and zu2004≪u2004R.
We us
Charged wire in half circleA wire carrying a uniform line charge density λu2004>u20040 has been folded as
shown in the figure. Calculate the electric field at the point O and the
potential with V(∞)u2004=u20040.
The potential at the point O is obtained by decomposing the
contributions of the two segments and of the half circle. With V(∞=0) we
have (be careful at the order of limits of the integrales):
\frac{1}{{4\pi \varepsilon _0 }}\left(~ {\int\limits_
Hole-in-the-tableA horizontal table has a hole that allows a thread of length L to pass
without friction. This thread connects two blocks: 1) a block of mass m
that slides without friction on the table and whose initial velocity is
not in the thread’s direction and 2) a block of mass M that’s vertically
hanging below the table. The thread is always taut.
-
Write the blocks’ total mechanical energy, considering that the
thread has constant length.
-
What are the conserved quantities, that is, the primary integrals
Wheel pulled by a blockA homogeneous, full, vertical wheel, of mass m and radius R, rolls
without slipping on a horizontal table. It is pulled at axis level by an
inextensible thread going through a pulley. At the other end of the
thread is suspended a block of mass m (see diagram). The thread and the
pulley have negligible masses, the thread remains taut at all times and
the system is initially at rest. After some time, the wheel has moved a
distance d forwards.
-
What is, then, the velocity of its center of mass ?
-
Given
Pendulum with a spring twistA material point of mass m is restricted to move without friction along
a vertical circle of radius R. It is subjected to two forces: its
weight, and the force from a spring linking it to a fixed point F on the
circle, at the same height as the circle’s center. The spring has spring
constant k and null rest length.
-
Show that the material’s point potential energy as a function of θ
is
E_(pot)(θ)u2004=u2004u2005−u2005kR²cosu2006θu2005−u2005mgRsinu2006θu2005+u2005constant.
-
Show that the two e
Coil with alternative currentIn a coil of radius R, with N turns per meter and of infinite length, an
alternative current circulates I(t)u2004=u2004I_(max)u2006cosu2006ωt.
Determine the amplitude of the electric field induced at a distance
ru2004>u2004R from the axis of the coil.
Same question but this time inside of the coil(ru2004
