Download Tutoring Session 11: Motional EMF and Work Done by

Electricity and Magnetism: PHY-204
Fall Semester 2014
Tutoring Session 11: Motional EMF and Work Done by
Magnetic Field (Part 2)
1. Consider a rectangular loop of wire of mass mw and length a suspended partially in
a magnetic field as shown below. The magnetic field exists only above the dashed
line and points into the plane of the paper. Throughout this question we assume that
any forces acting on the vertical sections of the loop are negligible compared to those
acting on the horizontal section.
The loop has current I running through it.
(a) Determine the current I0 at which the loop is in equilibrium.
(b) When I > I0 the loop ascends through a height h at constant acceleration. What
is the apparent work done by the magnetic force acting on the loop?
Given that a uniform magnetic field cannot do work, the answer to part (b) seems
like a contradiction. This apparent paradox can be resolved by showing that this
work comes from the power supply. This is what we now set to explore.
(c) Now consider the loop to be a superposition of fixed positive charges (ions) and
mobile negative charges (electrons). You are given below the free-body diagrams
for these positive and negative charge distributions, while the loop is moving with
velocity v(t). The total mass of the ion and electron charge distribution is M
and m, respectively. We have ignored resistive forces in the wire and forces due
to the battery in drawing these free body diagrams.
Date: 2 December, 2014
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Electricity and Magnetism: PHY-204
Fall Semester 2014
i. The positive and negative charge distributions exert an electric force (F ) on
each other. What is the source of this electric force?
ii. Reassure yourself that the total force acting on both the charged distributions
agrees with that calculated in part (a). You are given that the linear charge
density of the conduction electrons is λ.
iii. In what direction is the force from the power supply acting on both the
charge distributions?
−
iv. Find expressions for Fmag
and F .
(d) Given that the loop starts from rest and rises through a height ∆h in time ∆t,
i. Calculate the velocity of the loop after ∆t.
ii. Calculate the total work done by the electric and magnetic forces acting on
the two charge distributions as shown in the free-body diagrams of part (c).
(e) Using Ohm’s Law, show that
ε(t) − av(t)B = IR
where ε(t) is the external emf provided by the battery and R is the resistance of
the loop of wire.
(f) What is the total work done by the power supply? Hence, show that all the work
calculated in (b) comes from the power supply.
(g) What is the role of the magnetic forces in raising the loop of wire?
Date: 2 December, 2014
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Electricity and Magnetism: PHY-204
Fall Semester 2014
2. A conducting rod of length l moves with a constant velocity in a direction perpendicular to a long, straight wire carrying a steady current I, as shown below. Find the
magnitude of the emf generated between the ends of the rod.
The same rod now moves with a velocity v parallel to the wire. The axis of the rod is
maintained perpendicular to the wire with the near end a distance r away, as shown
below. Find the magnitude of the emf induced.
3. A bar of mass m and resistance R slides without friction in a horizontal plane, moving
on parallel rails as shown overleaf. The rails are separated by a distance d. A battery
that maintains a constant emf ε is connected between the rails, and a constant magnetic
⃗ is directed perpendicularly out of the page. Assuming the bar starts from rest
field B
at time t = 0, find the speed v(t) with which it moves at time t.
Date: 2 December, 2014
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Electricity and Magnetism: PHY-204
Fall Semester 2014
4. A conducting rod of length l is free to slide on two parallel conducting bars as shown
below. Two resistors R1 and R2 are connected across the ends of the bars to form
a loop. A constant magnetic field B is directed perpendicularly into the page. An
external agent pulls the rod to the left with a constant speed v.
Find
(a) the currents in both resistors,
(b) the total power delivered to the resistance of the circuit, and
(c) the magnitude of the applied force that is needed to move the rod with this
constant velocity. What is the power delivered by this force?
Date: 2 December, 2014
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