Download B for a circular coil

EXPERIMENT 9: THE MAGNETIC FIELD OF A CIRCULAR COIL
Object: To investigate the behavior of a magnetic field along the axis of a circular coil.
Prior to Lab: Solve equation 1 below for the formula of B at x = 0 Call it B(0). Write a formula
for the ratio of the field for any position x to B(0). Label it B(x)/B(0). For x = 0 to x = 25 cm
in 2.5 cm steps, then using Excel create the graph of B(x)/B(0). Assume a = 12.5 cm.
THEORY:
The magnetic field due to a circular coil is given by
B=
μ oN pI
3
2⎤2
(Eqn 1)
⎡ ⎛ x⎞
2a ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ a ⎠ ⎦⎥
where Np is the number of turns of the (primary) coil, a is the coil’s radius, and I is the
current in the coil.
To detect the magnetic field, one must can use Faraday’s Law if the magnetic field is
changing. A time dependent current will generate a changing magnetic field in Eqn 1. A
search coil may then be employed to intercept the time dependent flux and generate an emf
according to Faraday’s Law. The derivation proceeds as follows:
A transformer is used to provide a current according to I = Io sin 2πft, thus the field
becomes
B=
μ o N pI o sin 2πft
3
2⎤2
(Eqn 2)
⎡ ⎛ x⎞
2a ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ a ⎠ ⎥⎦
where t is time in seconds, and f is the frequency of the current variation. The magnetic
flux may then be calculated as
Φ =
∫ BdA
= BA sc
(Eqn3)
where Asc is the effective area of the search coil.
The flux is then
Φ=
μ o N pI o sin 2πft
3
2⎤2
A sc
(Eqn4)
⎡ ⎛ x⎞
2a ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ a ⎠ ⎥⎦
Inserting Eqn 4 into Faraday’s Law yields
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1
ε = −Nsc dΦ
dt
= − Nsc
⎧
⎫
⎪
⎪
⎪
⎪
⎪
d ⎪ μ o N p I o sin 2πft
A sc ⎬
⎨
3
dt ⎪
⎪
2⎤2
⎡
⎪ 2a ⎢1 + ⎛ x ⎞ ⎥
⎪
⎜ ⎟
⎪
⎪
⎝
⎠
a
⎢⎣
⎥⎦
⎩
⎭
where Nsc is the number of turns of the search coil and which upon evaluation yields
ε = −Nsc
μ o N p I o 2πf cos 2πft
3
2⎤2
(Eqn 5)
A sc
⎡ ⎛ x⎞
2a ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ a ⎠ ⎥⎦
π
Notice that the equation shows that the emf is time dependent and out of phase by 2
with the current.
The method of measurement is to measure the rms or effective values of the current
and the emf. By doing this the time dependent terms may be averaged out of the
equations, and the magnitudes of the values are all that is required. Thus Eqn 5 can be
reduced to
μ o NI
ε = 2πfNsc
3
2⎤2
(Eqn 6)
A sc
⎡ ⎛ x⎞
2a ⎢1 + ⎜ ⎟ ⎥
⎢⎣ ⎝ a ⎠ ⎥⎦
Upon substituting B for the expression for B embedded in Eqn 6 the equation simplifies
to
ε = 2πfNsc AscB
(Eqn 7)
which upon solving for B becomes
B=
By measuring
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ε
2πfNsc A sc
(Eqn 8)
ε one can then experimentally determine B from Eqn 8.
2
ACV
20Ω
Primary
coil
Search
coil
ACV
AC
door bell
transformer
Figure 1. The experimental apparatus.
Figure 2. The wiring diagram for the apparatus
shown in Fig. 1
PROCEDURE:
1. Wire the apparatus according to the above figure. DO NOT PLUG IN TRANSFORMER.
Do ask your instructor to check your circuit for correctness of wiring.
2. Record all the constants of the apparatus needed for calculations.
3. Set the DMM to 20V AC. Plug in the transformer and measure the voltage across the 20
Ω resistor.
4. Connect the DMM to the search coil leads, change the range of the DMM to give readings
of at least 3 significant figures, and record the induced voltage across the search coil for
each 2.5 cm.
ANALYSIS:
1. In the ACV mode, the uncertainty of the reading for the DMM is 1.5% of reading plus 4
digits.
2. Set up a spread sheet data table and calculate the theoretical magnetic field (Eqn 1) as
a function of position. Graph these values as a line. NOTE: The radius (a) of the
primary coil is 12.5 cm and the frequency of the AC source is 60 Hz.
3. Using the same spreadsheet, calculate B determined from your measured values of
and Eqn 8. Graph these values of B as symbols on the same graph as in step 2.
ε
4. Determine uncertainties of B for Eqn 8 for x = 0,15 cm and 25 cm using the instrument
uncertainty of the DMM to determine the uncertainty of the incuded emf, ΔNsc = 1, and
ΔAsc = 0.05 cm2.
REPORT:
Your report should include a discussion on the nature of the characteristic curve for B, any
discrepancies between measured and theoretical values, and sources of error.
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