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Consider a coxial cable of infinite length through which we put a
continuous current I. The current propagates in a direction via the
central conductor and comes back in the opposite direction via the
external conductor. We will suppose that at each instant the currents
are constant and uniform on the sections of the conductors and that
χ_(m)u2004=u20040 everywhere. Calculate the magnetic field in each point of
space and give in a graphical representation.By the cylindrical symmetry of the problem, the ma
Two circular coils of n turns for each and of radius R are centered on
the x axis. The thickness of the coils is negligible. The center of the
first coil is in xu2004=u20040 and the one of the second in xu2004=u2004R. A constant
current I circulates in the same direction in each coil. Show that the
magnetic field on the x axis is :
B=2Nμ0IR2[(x2+R2)231+(2R2+x2−2xR)231]Because of the symmetry of rotation around the x axis, the compo
In 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
Calculate, in any point of space, the electrostatic potential
V(r) and the field E(r) created by a spherical shell
uniformly charged with an internal radius a, external b, and of total
charge u2005+u2005Q, centered at the origin O of the cartesian axes. Graphically
represent the functions V(r) and Er(r) (the radial
component of E(r))Replace now the charged shell by an ideal conductor of similar shape.
The conductor is neutral and isolated. A charge u2005+u2005
A closed wire is made up with two semicircles and two straight wires as
shown in Fig. 1. The two semicircles have radii 2R and R. The straight
wires connect the two semicircles and are placed in the diameter’s
direction. A current I is circulating through the wire. Calculate the
magnetic field at the centre of the semicircles.[A closed wire made up of two semicircles..]
The magnetic field is
dB=4πr2μ0Idl×r^. Since the
two straight wires pass through the center, $
Consider 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
A ladder with mass m and length L is resting on a vertical wall. The
angle between the wall and the ladder is α and the friction coefficient
between the ladder’s foot and the floor isxa0μ. There is no friction
between the wall and the ladder.What’s the maximal value αmax angle α can take such that
the ladder’s bottom doesn’t slip on the floor and fall?
We consider the ladder as the system, seen from the referential of the
floor and the wall. We choose a Cartesian reference frame with a
horiz
The surface of the cylinder of radius R and of length hu2004≫u2004R is uniformly
charged with a charge density σ. What is the magnetic field inside the
cylinder when it rotates around its axis at an angular velocity ω?
N.A.: Ru2004=u20040.1xa0m; σu2004=u20041u2006Cm⁻²; ωu2004=u20042πu2005×u200510³u2006s⁻¹
The rotation of the charged cylinder generates a current. To determine
this current, let us consider a rectangular element of surface with
dimensions : dl (basis) and h (height). The charge carried by this
A 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_
A solide is made up of a truncated straight homogeneous cone, with a
circular base of radius R, set on a homogenous half-sphere of radius R
and with the same density ρ, as shown on the diagram. The cone opens
downwards with an apex half-angle α.Show that the solid is in a stable equilibrium on the horizontal plane
if, and only if αu2004>u200430^(o).Hint: With origin O at the junction between both solids on axis z, the
center of mass of the cone with volume 31πhR2 is at
height $z_{G_c}=\fr
An incompressible liquid circulates in a tube. At a certain location,
the section of the tube strongly decreases (cf. figure). The continuity
equation indicates us that :
∂t∂ρ+∇⋅ρv=0.
Show that this is equivalent to the simpler equation:\quad \langle v_n\rangle=\frac{1}{S_n}\iint_{S_n}\vec{v}\cdot d\vec{\sigma} ~.$$
If we add to the figure the velocity vectors as well as the vect
A wire of length L and of constant line charge density λ is vertically
aligned between z=−2L and z=2L. Calculate the
electric field E at point Pu2004=u2004(r,0,0). Check the limits Lu2004≫u2004r and
Lu2004≪u2004r.
By the symmetry of the problem, the created field in P(r,0,0) is radial:
E=Err^. The contribution of an infinitesimal element dz
of charge dq at point P(r,0,0) is:
$$dE_{r}=\frac{1}{4 \pi \epsilon_{0}}\frac{dq}{{\mbox{$\resizebox{.16in}{.08in}{\includeg
A yoyo is made up of a disk of radius R and mass M, around which a
massless thread is wound up. The thread is attached to the ceiling and
remains constantly vertical; gravity acts on the yo-yo which is free to
lower down.
Compute the disk’s moment of inertia I around an axis perpendicular
to the disk’s plane and going through its center.
Find the acceleration a⃗ of the yo-yo’s center of mass, as well as
the thread’s tension T⃗.
A disk’s moment of inertia with respect to its revolution
Consider 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:
xa0Determine the position in the plane of the center of mass of this system
consisting of the three square plates, as represented of the figure
hereby. The square plates are homogeneous and their edges have a length
a.The two light plates have the same mass m, the mass of the dark plate
is the double.xa0We are in the coordinate system (O,e_(x),e_(y)) of the sketch hereby. By
symmetry, the centre of mass of each square plate is at its centre : so
at the points
$\left(\frac{a}{2}, \frac{a}{2}\right) , \
A mass m, subject to gravity, is hanging on a rectangular circuit in
which circulates a current I. The superior part of the circuit is
located inside of an uniform magnetic field B₀ perpendicular to the
plane of the figure.Determine B₀ such that the mass freely floats.
When we increase the current, we notice that the circuit and the mass
are moving upwards. What is the work realised by the magnetic field when
the mass goes upwars with Δzu2004=u2004h?The field applies on the horizontal side a force
${\bf
An infinite cylinder of ferromagnetic material carries a uniform
magnetization M. What is the current density linked to the
surface of the cylinder? Determine the fields B and H
inside and outside of the cylinder.
At the surface of the magnetized cylinder a linked current whose
density(current per unit length) is Kb=Mϕ^ is
circulating. The surface of the cylinder is behaving like an infinite
coil that carries the same current density. So B=H=0
o
xa0Determine the position of the centre of mass of the quarter of a full
sphere of radius R and of constant volumic mass.xa0We describe the quarter of sphere by using the spherical coordinates
such that the planes (Oxy) and (Oyz) are the two planes of symmetry. The
centre of mass is in the line situated on the two planes, so (Oy). The
calculation of y_(cm), the projection of the centre of mass on Oy, is
enough to define the position of the centre of mass.
The projection of r⃗ on Oy is given by
$y = \v
xa0An attraction sometimes seen in fairs consists in a motor-biker entering
in a spherical cage and starting to circularly drive faster and faster.
At the start of the rotation, the motor-biker is located at the bottom
of the sphere, then, as the velocity increases, he goes up. As such, he
can reach the medium level of the sphere. In this situation, le body of
the motor-biker is horizontal(αu2004=u200490^(∘)).
Let us consider a spherical cage of radius R, and a motor-biker (on its
motorcycle) seen as a m
A 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
