EMI 3c: Circular Loops within a Solenoid ................................................................... 2
EMI3c—RT1: Circular Loop Within a Solenoid ............................................................................................. 3
EMI3c—WBT1: Circular Loop Within a Solenoid .......................................................................................... 4
EMI3c—WWT1: Circular Loop Within a Solenoid ......................................................................................... 5
EMI3c—TT1: Circular Loop Within a Solenoid.............................................................................................. 6
EMI3c—CCT1: Circular Loop Within a Solenoid........................................................................................... 7
EMI3c—CRT1: Circular Loop Within a Solenoid........................................................................................... 8
EMI3c—PET1: Circular Loop Within a Solenoid ........................................................................................... 9
EMI3c—QRT1: Circular Loop Within a Solenoid......................................................................................... 10
EMI3c—M/MCT1: Circular Loop Within a Solenoid .................................................................................... 11
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EMI 3C: CIRCULAR LOOPS WITHIN A SOLENOID
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EMI3C—RT1: CIRCULAR LOOP WITHIN A SOLENOID
The figures below show six situations where circular loops of wire are placed inside six
solenoids. The solenoids, which have uniform fields in their interiors, are concentric with the
loops but have larger diameters. The circular loops vary in diameter, which of course means they
have different areas. We are told the areas for the loops as well as the INDUCED current in each
loop as a result of a change in the strength of the field within the solenoid. (Assume all loops
have the same resistance.)
Rank these situations, from greatest to least, on the basis of the change in current per unit
time in the solenoids that produced the induced currents in the circular loops.
A
B
C
A = 24 cm2
A = 24 cm2
A = 32 cm2
I = 8 mA
I = 6 mA
I = 8 mA
D
E
F
A = 16 cm2
A = 32 cm2
A = 16 cm2
I = 8 mA
I = 6 mA
I = 4 mA
Greatest
1 ___D__ 2 ___A__ 3 _B C F_ 4 ______ 5 ______ 6 ___E___
Least
OR, The change in current per unit time is the same for ALL SIX solenoids. _______
OR, There is no change in current in any of the solenoids. _______
Please carefully explain your reasoning.
In each case, the area of the loop is fixed; thus, the change of flux is due to a change
in the magnetic field. The change in the magnetic field is due to a change in current in the
solenoid. Since we are giving the induced currents, the change in current per unit time in the
solenoid is the induced current divided by the area or
Iloop =
Emfinduced
dφ d ( B ⋅ Aloop )
dB
di
di
∝ Emfinduced =
=
= Aloop ⋅
= Aloop ⋅ µ0 ⋅ nsolenoid ⋅ solenoid ∝ Aloop ⋅ solenoid
Rloop
dt
dt
dt
dt
dt
I
disolenoid
∝ loop since the B = µ0 ⋅ n ⋅ isolenoid
dt
Aloop
How sure were you of your ranking? (circle one)
Basically Guessed
Sure
1
2
3
4
5
6
7
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8
9
Very Sure
10
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EMI3C—WBT1: CIRCULAR LOOP WITHIN A SOLENOID
Construct a physical situation to which the following relation could apply.
12 mA =
turns
)( I2 − I1 )
1
cm
(π )( 4 cm)2
3 msec
6Ω
( µo )(800
Comparing it to the equation for the current induced in a loop placed in within the magnetic
field of a solenoid that is changing the currentIloop =
∆i
Emfinduced
= Aloop ⋅ µ0 ⋅ nsolenoid ⋅ solenoid
∆t
Rloop
Rloop = µ0 ⋅ nsolenoid ⋅
∆isolenoid
2
⋅π ⋅ rloop
⋅ Rloop
∆t
We see that one physical situation that could be constructed is to place a circular loop with
radius of 4 cm and a resistance of 6 ohms inside a solenoid such that the centers are
concentric and aligned parallel. The solenoid has 800 turns per centimeter and the current is
changing from I1 to I2 in 3 milliseconds.
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EMI3C—WWT1: CIRCULAR LOOP WITHIN A SOLENOID
What, if anything, is wrong with the following situation?
A circular wire loop that is inside, and concentric with, a solenoid will have an induced
current if the current in the solenoid changes.
Looks ok.
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EMI3C—TT1: CIRCULAR LOOP WITHIN A SOLENOID
Something is wrong with the situation described below. Identify the problem and explain how
to correct it.
A circular wire loop that is inside, and concentric with, a solenoid has an induced current
in it. Consequently, we know that the current in the solenoid must be increasing.
This could be correct, but the changing current could also be decreasing.
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EMI3C—CCT1: CIRCULAR LOOP WITHIN A SOLENOID
The statements below are about a situation where a circular wire loop is inside and concentric
with a large solenoid.
Student A: “If the current in the solenoid changes, there will be an induced current in the wire
loop because the magnetic field will change.”
Student B: “Changing the current in the solenoid will not produce an induced current since
the magnetic field will still be uniform.”
Student C: “Changing the current in the solenoid will have no effect on any current in the
wire if they are not connected together.”
With which, if any, of these students do you agree?
Student A __X__
Student B ______
Student C_______
Disagree with all_______
Carefully explain your reasoning.
The magnetic field of the solenoid produces a magnetic flux through the loop. If the current in
the solenoid changes, this changes the magnetic flux through the loop. This magnetic flux
change produces an induced emf that produces a current in the loop.
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EMI3C—CRT1: CIRCULAR LOOP WITHIN A SOLENOID
The graph below shows how the current in a large solenoid of length L, diameter D, and n turns
per unit length changes during a certain time interval. Write the specific expression(s) for the
induced current in a circular wire loop of resistance R and diameter d, which is inside the
solenoid, concentric with it, and at the center of the solenoid.
Current
(A)
I3
I2
I1
0
Time
(s)
0 to t1: (assuming that that the entire loop is within the solenoid or d<D)
0
Iloop =
t1
t2
t3
Emfinduced
I −I
d2
∆i
2
= µ0 ⋅ nsolenoid ⋅ solenoid ⋅π ⋅ rloop
⋅ Rloop == µ0 ⋅ nsolenoid ⋅ 2 1 ⋅π ⋅ ⋅ Rloop
Rloop
t1 − 0
∆t
4
t1 to t2:
Iloop =
Emfinduced
= 0 since there is no change in current in the solenoid
Rloop
t2 to t3: (assuming that that the entire loop is within the solenoid or d<D)
Iloop =
Emfinduced
I −I
d2
∆i
2
= µ0 ⋅ nsolenoid ⋅ solenoid ⋅π ⋅ rloop
⋅ Rloop == µ0 ⋅ nsolenoid ⋅ 3 2 ⋅π ⋅ ⋅ Rloop
Rloop
t3 − t2
∆t
4
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EMI3C—PET1: CIRCULAR LOOP WITHIN A SOLENOID
A solenoid with a large diameter has a current I1 in it. A circular wire loop of smaller diameter is
placed inside the solenoid with the axis of the loop parallel to the axis of the solenoid. The wire
loop is connected to an ammeter. The current in the solenoid increases to three times its initial
value during a 50-millisecond time interval.
What will we observe on the ammeter? Explain.
A steady or constant current will be observed since the induced emf will be constant due to the
constant rate of change of the current in the solenoid.
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EMI3C—QRT1: CIRCULAR LOOP WITHIN A SOLENOID
A wire loop is placed inside and concentric with a large diameter solenoid. The solenoid has an
initial current I in it. The initial current decreases to half in a time interval ∆t. This change will
produce an induced current IInd in the wire loop.
Explain how each of the changes described below would affect the induced current in the
wire loop.
The basic equation for this situation is Iloop =
∆i
Emfinduced
= Aloop ⋅ µ0 ⋅ nsolenoid ⋅ solenoid
∆t
Rloop
Rloop
The solenoid current decrease occurs in a longer time than the time above.
Smaller since delta t is longer.
The diameter of the wire loop is doubled but the loop is still inside the solenoid.
Larger by a factor of two since the area increases by a factor of four while the resistance
increases by a factor of two so the ratio is the factor of two.
The diameter of the solenoid is increased.
No difference since the magnetic field is the same.
The current decrease in the solenoid occurs in less than the original time interval.
Larger since the time interval is shorter.
The initial solenoid current is three times the value above but decreases to half in the same time
interval.
∆i
Emfinduced
The basic equation for the situation Iloop =
= Aloop ⋅ µ0 ⋅ nsolenoid ⋅ solenoid Rloop
∆t
Rloop
3I
3I
Everything is the same except ∆isolenoid = 3 I0 − 0 = 0 so the induced current is larger by a
2
2
factor or 3 (the initial situation gave a change of 1/2)
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EMI3C—M/MCT1: CIRCULAR LOOP WITHIN A SOLENOID
A student is working with a situation where a solenoid (n = 120 turns / m) with a large radius, r =
12 cm, has a current of 4 A in it. A circular wire loop of smaller radius, r = 5 cm, is placed inside
the solenoid with the axis of the loop parallel to the axis of the solenoid. The wire loop is
connected to an ammeter. The current in the solenoid increases to three times its initial value
during a 50-millisecond time interval. The resistance of the wire loop is 15 ohms. The student
performs the calculation below for the induced current in the loop.
Is this calculation meaningful or meaningless?
I=
(µo )(120m-1 )(12 A − 4 A)(π )(.05m) 2
(5 × 10 -3 s)(15Ω)
Explain fully.
Meaningful since the basic equation is
∆i
Emfinduced
Iloop =
= Aloop ⋅ µ0 ⋅ nsolenoid ⋅ solenoid
∆t
Rloop
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