Description
Problem 1. A particle with mass m and charge q moves in mutually perpendicular electric and magnetic
fields E = (0, 0, E0) and B = (B0, 0, 0), where E0 and B0 are positive constants. Find and
sketch the trajectory of the particle if it starts at the origin with velocity
(a) v(0) = (E/B)ˆny,
(b) v(0) = (E/2B)ˆny,
(c) v(0) = (E/B)(ˆny + ˆnz).
You are allowed to use all results we have derived in class.
(3 × 3/2 marks)
Problem 2. A particle with mass m and positive charge q moves in antiparallel electric and magnetic fields
E = (−E0, 0, 0) and B = (B0, 0, 0), where E0 and B0 are positive constants. Assuming the initial
conditions: v(0) = (v0x, v0y, 0) and r(0) = (0, 0, 0), find the velocity v(t) and position r(t) for
t > 0.
(6 marks)
Problem 3. A plane wire loop of irregular shape is situated so that part
of it is in a uniform magnetic field B (in the figure below the
field occupies the shaded region and points perpendicular to
the plane of the loop). The loop carries the current I. Show
that the magnitude of the net magnetic force on the loop is
F = IBw, where w is the chord subtended.
What is the direction of the force?
(3 marks)
Problem 4. Magnetic forces acting on conducting fluids provide a convenient means of pumping these fluids. For example, this method
can be used to pump blood without the damage to the cells that
can be caused by a mechanical pump. A horizontal tube with
rectangular cross section (height h, width w) is placed at right
angles to a uniform magnetic field with magnitude B so that a
length l is in the field (see the figure below). The tube is filled
with a conducting liquid, and an electric current of density J
is maintained in the third mutually perpendicular direction.
(a) Show that the difference of pressure between a point in
the liquid on a vertical plane through ab and a point in
the liquid on another vertical plane through cd, under
conditions in which the liquid is prevented from flowing,
is ∆p = JlB.
(b) What current density is needed to provide a pressure difference of 1 atm between these two points if B = 2.2 T
and l = 35 mm?
(2 + 1 marks)
Problem 5. In class we derived an expression for the torque on a current loop assuming that the magnetic
field B was uniform. But what if B is not uniform?
Assume we have a square loop of wire that lies in the xy–plane. The loop has corners at (0, 0),
(0, L), (L, L) and (L, 0) and carries a constant current I in the clockwise direction. The magnetic
field B = (B0y/L, B0x/L, 0), where B0 is a positive constant.
(a) Sketch the magnetic field lines in the xy–plane.
(b) Find the magnitude and direction of the magnetic force exerted on each of the sides of the
loop.
(c) If the loop is free to rotate about the x–axis, find the magnitude and direction of the
magnetic torque on the loop.
(d) Repeat part (c) for the case in which the loop is free to rotate about the y–axis.
(e) Is equation τ = µ × B an appropriate description of the torque on this loop. Why or why
not?
(1/2 + 1 + 2 + 2 + 1 marks)
Problem 6. A circular loop of radius R carries a clockwise electric current I. The
loop is placed in a uniform magnetic field B (see the figure).
(a) What is the net force on the current loop?
(b) Find the torque on the current loop with respect to the axis of
symmetry of the loop perpendicular to the vector B.
(1 + 3 marks)
Problem 7. In a certain region of space, the magnetic field B is not uniform: it has both a z–component and
a component that points radially away from or towards the z–axis. The z–component is given
by Bz(z) = βz, where β is a positive constant. The radial component Br depends only on r, the
radial distance from the z–axis. (a) Use Gauss’s law for magnetism, to find Br as a function of
r. (b) Sketch the magnetic field lines.
(2 + 1 marks)
