Matthew Feickert
PHYS 1304: Introductory Electricity and Magnetism
Quiz 7 Key
March 31th, 2015
Problems
1. (6 points) A particle with positive charge q = 4.00 × 10−18 C moves with a velocity of v = 2.50 × 106 m/s (− ŷ) perpendicularly toward a long straight wire with current
I = 800 mA flowing along x̂. At a certain instant in time, the particle’s radial distance from
the wire is d = 4.00 cm. What is the magnitude and direction of the force on the particle
due to the current?
2. (4 points) Check all the statements that are correct.
(a) A beam of moving protons generates a magnetic field around it.
(b) The force between two anti-parallel currents is attractive.
(c) The magnetic field inside a toroid is uniform.
(d) The magnetic field lines inside a solenoid go from the south pole to the north pole.
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Solutions
1. (6 points) A particle with positive charge q = 4.00 × 10−18 C moves with a velocity of v = 2.50 × 106 m/s (− ŷ) perpendicularly toward a long straight wire with current
I = 800 mA flowing along x̂. At a certain instant in time, the particle’s radial distance from
the wire is d = 4.00 cm. What is the magnitude and direction of the force on the particle
due to the current?
From Ampère’s law,
∫
B · dl = µ0 Ienc ,
it is seen that for a circular Amperian loop of radius r,
B (2πr) = µ0 I,
that at a radial distance of r from the wire, the magnetic field experienced by the particle—
according to the “right hand rule”—is
µ0 I
ẑ.
B=
2πr
From the Lorentz force law,
F = q (E + v × B) ,
it is seen that at a radial distance d, the particle experiences a force of
F = qv × B
µ0 I
= qv
(−ŷ × ẑ)
2πd
q vµ0 I
= −
x̂
2πd
(
)
(4.00 × 10−18 C) (2.50 × 106 m/s) 4π × 10−7 V · s · A−1 · m−1 (0.8 A)
=−
x̂
2π (0.04 m)
= −4 × 10−17 N x̂.
Points:
• 2 points for correct expression of magnetic field magnitude, |B|
– If they have incorrect B, but used Ampère’s law they receive 1 point
• 1 point for Lorentz force law (qv × B)
• 2 point for correct algebraic expression of magnitude (q vµ0 I/2πd)
• 0.5 points for correct numeric magnitude (4 × 10−17 N)
• 0.5 points for correct direction (−x̂)
• -1 point for wrong units
2
2. (4 points) Check all the statements that are correct.
(a) A beam of moving protons generates a magnetic field around it. ✓
Current is moving charges,
dQ
,
dt
and it is known from Ampère’s law that current produces a magnetic field.
I=
(b) The force between two anti-parallel currents is attractive. 7
Use the “right hand rule” and see that the force is repulsive.
(c) The magnetic field inside a toroid is uniform. 7
Btoroid =
µ0 IN
,
2πr
a≤r≤b
(d) The magnetic field lines inside a solenoid go from the south pole to the north pole. 7
Magnetic field lines go from the north pole to the south pole.
Points:
• 1 point for each correctly checked box (✓).
• 1 point for each incorrect box left blank (7).
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Alternative Solutions
1. (6 points) A particle with positive charge q = 4.00 × 10−18 C moves with a velocity of v = 2.50 × 106 m/s (− ŷ) perpendicularly toward a long straight wire with current
I = 800 mA flowing along x̂. At a certain instant in time, the particle’s radial distance from
the wire is d = 4.00 cm. What is the magnitude and direction of the force on the particle
due to the current?
From Ampère’s law,
∫
B · dl = µ0 Ienc ,
it is seen that for a circular Amperian loop of radius r,
B (2πr) = µ0 I,
that at a radial distance of r from the wire, the magnetic field is
B=
µ0 I
ϕ̂.
2πr
From the Lorentz force law,
F = q (E + v × B) ,
it is seen that—switching to cylindrical coordinates (r, ϕ, z) with ẑ being along the direction
of the current—at a radial distance d, the particle experiences a force of
F = qv × B
)
µ0 I (
= qv
−r̂ × ϕ̂
2πd
q vµ0 I
q vµ0 I
= −
ẑ cylind = −
x̂cart .
2πd
2πd
4