Download Week 8 - Magnetic Field and Magnetic Forces

Week 8 - Magnetic Field and Magnetic Forces
Ampere was the Newton of Electricity.
James C. Maxwell
Exercise 8.1: Discussion Questions
a) Can a charged particle move trough a magnetic field without experiencing any force? If so, how? If
not, why not?
b) If the magnetic field do no work, how can it have a significant effect on a particles motion? Are there
any other examples for which a force has a significant effect on the motion without doing any work?
c) A student claims that if lightning strikes a metal flag pole, the force exerted by the earths magnetic
field on the current flowing trough the pole can be enough to bend it. Typical lightning currents are
of the order of 104 to 105 amperes. Is the student’s opinion justified?
d) A loudspeaker should never be placed in next to a computer monitor or a TV screen. Why not?
Exercise 8.2: Gauss’ Law for Magnetism
In a certain region of space, the magnetic field B is not uniform. The magnetic field has both a zcomponent and a component that points radially away from the z-axis. The z-component is given by
Bz (z) = βz, where β > 0. The radial component Br depends only on r, the radial distance from the
z-axis.
a) Use Gauss’ law for magnetism to find the radial component Br as a function of r. Hint: Try a
clindrical Gaussian surface of radius r concentric with the z-axis with one end at z = z0 and the
other at z.
Answer:
1
Br = − βr.
2
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Exercise 8.3: Mass Spectrograph
A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses. In
one design for such an instrument, ions with mass m and charge q are accelerated trough a potential
difference V . They then enter a uniform magnetic field that is perpendicular to their velocity, and
they are deflected in a semicicular path of radius R. A detector measures where the ions complete the
semicircle and from this it is easy to cacluate R. The situation where a source S is emitting ions is shown
1
Figure 1
in figure 1. In the red region there is a potential difference, while in the blue region there is a uniform
magnetic field.
a) Derive the equation for calculating the mass of the ion from measurements of B, V , R and q.
Answer:
m=
qB 2 R2
.
2V
b) What potential difference V is needed so that singly ionized
0.150 T magnetic field?
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12
C atoms will have R = 50.0 cm in a
c) Suppose the beam consists of a mixture of 12 C and 14 C ions. If V and B have the same values as in
part (b), calculate the separation of these two isotopes at the detector. Do you think that this beam
separation is sufficient for the two ions to be distinguished?
Answer:
∆R = 4 cm
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.
Figure 2
Week 8 – October 24, 2011
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compiled October 9, 2013
Exercise 8.4: An Electromagnetic Rail Gun
A conducting bar with mass m and length L slides over horizontal rails that are connected to a voltage
source. The voltage source maintains a constant current I in the rails and bar, and a constant, uniform,
vertical magnetic field B fills the region between the rails. This is shown in figure 2 where the black
arrows indicate the direction of the current trough the grey rails and the purple conducting bar, while
the blue arrows (pointing out of the page) indicate the direction of the B-field.
a) Find the magnitude and direction of the net force on the conducting bar. Ignore friction, air resistance,
and electrical resitance.
Answer:
FB = ILB î.
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b) If the bar has mass m, find the distance d that the bar must move along the rails from rest to attain
a speed v.
Answer:
d=
mv 2
.
2ILB
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c) It has been suggested that rail guns based on this prinicple could accelerate payloads into earth orbit
or beyond. Find hte distance the bar must travel along the rail if it is to reach the escape speed for
the eart (11.2 km/s). Let B = 1.0 T, I = 3.0 kA, m = 1.0 kg and L = 1.0 m. For simplicity assume
the net force on the object is equal to the magnetic force, as in parts (a) and (b), even though gravity
plays an important role in an actual launch into space.
Exercise 8.5: Magnetic Moment of the Hydrogen Atom
In the Bohr model of the hydrogen atom, in the lowest energy state the electron orbits the proton at a
speed of 2.2 × 106 m/s in a circular orbit of radius a0 = 5.3 × 10−11 m called the Bohr radius.
a) What is the orbital period of the electron?
Answer:
T =
2πr
.
v
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b) If the orbiting electron is considered to be a current loop, what is the current I?
Answer:
I=−
Week 8 – October 24, 2011
ev
2πr
3
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compiled October 9, 2013
c) What is the magnetic moment of the atom due to the motion of the electron? Show that it is
proportional to the orbital angular momentum of the electron.
Answer:
µ=−
eL
.
2me
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Exercise 8.6: Force on a Semicircular Wire
Figure 3 shows a conducting wire which enters a region of a uniform magnetic field B pointing out of
the page. In this region the wire takes the form of a semicirle. The direction of the current running
trough it is shown by the arrows. Find the net force on the conducting wire.
Figure 3
Answer:
F = −2IBRĵ.
Week 8 – October 24, 2011
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compiled October 9, 2013