Matthew Feickert
PHYS 1304: Introductory Electricity and Magnetism
Quiz 8 Key
April 7th, 2015
Problems
1. (6 points) A 120 turn coil of radius 10.0 cm and resistance 5.0 Ω is coaxial with a very
long solenoid of 240 turns/cm and diameter 4.8 cm. The current in the solenoid steadily
drops from 1.4 A to zero in time interval ∆t = 28 ms. What current (in Ampères) is induced
in the coil during ∆t?
2. (4 points) Check all the statements that are correct.
(a) The alternator in a car uses a rotating magnetic field to generate electricity.
(b) The alternator in a car uses a rotating electric field to generate electricity.
(c) A changing magnetic flux in space generates an electrical field.
(d) In a transformer the current in the secondary coil generates a magnetic field that
opposes the change of the magnetic field generated by the primary coil.
1
Solutions
1. (6 points) A 120 turn coil of radius 10.0 cm and resistance 5.0 Ω is coaxial with a very
long solenoid of 240 turns/cm and diameter 4.8 cm. The current in the solenoid steadily
drops from 1.4 A to zero in time interval ∆t = 28 ms. What current (in Ampères) is induced
in the coil during ∆t?
As the magnetic field of a solenoid for radius a and n windings per meter∗ is
{
µ0 nI ẑ, 0 ≤ r < a,
B sol =
0,
r > a,
the magnetic flux through a circular surface of radius b (b > a) around the solenoid is
I
ΦB = B · ds
(
)
= B πa2 .
Noting Faraday’s law of induction,
E = −N
dΦB
,
dt
it is seen by Ohm’s law that the electromotive force† (an electric potential) is
E = IR,
and so the current induced in the coil of N windings is
I′ =
=
=
=
≈
|E|
R
N dΦB
R dt
N ( 2 ) dB
πa
R
dt
µo nN πa2 dI
R
dt
µo nN πa2 ∆I
R
∆t
(1)
µo (24000 turns/m) (120 turns) π (0.024 m)2
(1.4 A)
(5 Ω)
(2.8 × 10−2 sec)
≈ 0.065 A.
≈
∗
†
Where ẑ is defined to be down the barrel of the solenoid.
Also called an “emf”
2
Points:
• 1 point for the correct magnitude of the solenoidal magnetic field (Bsol = µ0 nI)
• 1 point for Faraday’s induction law (|E| = N dΦB /dt)
– If Faraday’s induction law is wrong, but have correct expression for magnetic flux
award 0.5 points (ΦB = Bπa2 )
• 1 point for use of Ohm’s law (E = IR)
• 2 points for correct algebraic expression, Eq. (1)
• 1 point for correct numeric answer (≈ 0.065 A)
• -1 point for wrong units
3
2. (4 points) Check all the statements that are correct.
(a) The alternator in a car uses a rotating magnetic field to generate electricity. ✓
An alternator uses a rotating magnetic field to induce an AC voltage in the stator windings.
(b) The alternator in a car uses a rotating electric field to generate electricity. 7
An alternator uses a rotating magnetic field to induce an AC voltage in the stator windings.
(c) A changing magnetic flux in space generates an electrical field. ✓
By Faraday’s law of induction, a changing magnetic flux in space generates an electromotive force (emf) (an electric potential).
−
dΦB
=E
dt
I
= E · ds
(d) In a transformer the current in the secondary coil generates a magnetic field that
opposes the change of the magnetic field generated by the primary coil. ✓
This current, Is , produces a magnetic flux in the transformer’s core, which induces an
opposing emf in the primary coil.
Points:
• 1 point for each correctly checked box (✓).
• 1 point for each incorrect box left blank (7).
4